# First variation of functional Wulff shapes

Jacopo Ulivelli

###### Abstract

We introduce functional Wulff shapes based on the classical construction for
compact convex sets. With this new tool, we establish a functional version of
Aleksandrov’s variational lemma in the family of convex functions with compact
domain. The resulting formula is then applied to evaluate the first variation
of a class of functionals on convex functions. In particular, we extend a
recent result by Huang, Liu, Xi, and Zhao [26]. †† MSC 2020 Classification:
52A20, 52A38, 49J53.
Keywords: Convex function, convex body, Fenchel-Legendre transform, Wulff
shape.

##  1 Introduction

This paper follows in the footsteps of the so-called functionalization of
convex geometry. See, for example, [3, Chapter 9]. We focus on the family of
convex functions

$\mathrm{Conv}(\mathbb{R}^{n})\coloneq\\{u:\mathbb{R}^{n}\to\mathbb{R}\cup\\{+\infty\\}\colon
u\text{ is convex, lower semi-continuous, }u\not\equiv+\infty\\}.$

For $u\in\mathrm{Conv}(\mathbb{R}^{n})$, consider the functional volume

$\mu_{n}(u)\coloneqq\int_{\mathbb{R}^{n}}e^{-u(x)}\,dx.$ (1)

This choice is justified by the Prékopa-Leindler inequality (see, for example,
[21]) and the functional Blaschke-Santaló inequality [4], where $\mu_{n}$
replaces the volume in the corresponding classical inequalities for compact
convex sets. A different angle on measures of convex functions has been
pursued in the recently developing theory of valuations on convex functions
[11, 12, 13, 14, 15, 16, 24, 34, 35, 36, 37, 39, 43, 45].

Consider the family $\mathcal{K}^{n}$ of non-empty, compact, and convex
subsets of $\mathbb{R}^{n}$. We denote by $\mathrm{Vol}$ the $n$-dimensional
Lebesgue measure on $\mathbb{R}^{n}$, and write $K+L$ for the Minkowski sum of
for $K,L\in\mathcal{K}^{n}$. It is a classical fact (see, for example, [52,
(5.33)]) that the right derivative of $\mathrm{Vol}(K+tL)$ at $0$ takes the
explicit form

$\left.\frac{d}{dt}\mathrm{Vol}(K+tB^{n})\right|_{t=0^{+}}=\int_{\mathbb{S}^{n-1}}h_{L}\,dS_{K},$
(2)

where $\mathbb{S}^{n-1}$ is the Euclidean unit sphere, $h_{L}$ is the support
function of $L$, and $S_{K}$ is the surface area measure of $K$ (more on these
definitions in Section 2). A similar question can be posed in
$\mathrm{Conv}(\mathbb{R}^{n})$. In this case, $\mathrm{Vol}$ is replaced by
$\mu_{n}$, and Minkowski addition by infimal convolution, defined as

$u\square v(x)\coloneqq\inf_{y\in\mathbb{R}^{n}}\\{u(y)+v(x-y)\\}$

for $u,v\in\mathrm{Conv}(\mathbb{R}^{n})$. Colesanti and Fragalà [10]
investigated the first variation

$\left.\frac{d}{dt}\mu_{n}(u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v))\right|_{t=0^{+}}$
(3)

for $u,v\in\mathrm{Conv}(\mathbb{R}^{n})$ in a suitable class (see Section 2
for the definition of
$t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v$). The same authors
further explored the subject of functional mixed volumes together with Bobkov
[7]. Later, Cordero-Erasquin and Klartag [17] studied (3) in connection with
moment measures in the family of essentially continuous convex functions.
Rotem [50, Theorem 1.5] finally obtained a general formula for the first
variation (3), which reads as follows. For $u\in\mathrm{Conv}(\mathbb{R}^{n})$
we write $\mathrm{dom}(u)\coloneqq\\{x\in\mathbb{R}^{n}:u(x)<+\infty\\}$ and
we denote its gradient by $\nabla u$. For a convex set $K$, we denote its
Gauss map by $N_{K}$.

######  Theorem 1.1 (Rotem).

Let $u,v\in\mathrm{Conv}(\mathbb{R}^{n})$. If $0<\mu_{n}(u)<+\infty$, then

$\begin{gathered}\left.\frac{d}{dt}\mu_{n}(u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v))\right|_{t=0^{+}}\\\
=\int_{\mathrm{dom}(u)}v^{*}(\nabla
u(x))e^{-u(x)}\,dx+\int_{\partial\mathrm{dom}(u)}h_{\mathrm{dom}(v)}(N_{\mathrm{dom}(u)}(y))e^{-u(y)}\,d\mathcal{H}^{n-1}(y).\end{gathered}$
(4)

Equivalently, (4) can be rewritten as

$\left.\frac{d}{dt}\mu_{n}(u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v))\right|_{t=0^{+}}=\int_{\mathbb{R}^{n}}v^{*}(y)\,d\mathcal{M}_{u}(y)+\int_{\mathbb{S}^{n-1}}h_{\mathrm{dom}(v)}(\boldsymbol{\nu})\,d\mathcal{S}_{u}(\boldsymbol{\nu}),$
(5)

where $\mathcal{M}_{u}$ and $\mathcal{S}_{u}$ are the push-forward of the
measures with densities $e^{-u(x)}dx$ and $e^{-u(y)}d\mathcal{H}^{n-1}(y)$
obtained through the gradient of $u$ and the Gauss map of $\mathrm{dom}(u)$
respectively. This form has a striking parallelism with (2), which is
recovered if $u$ and $v$ are the indicator functions of
$K,L\in\mathcal{K}^{n}$.

More recently, Huang, Liu, Xi, and Zhao [26] investigated this problem in a
generalized setting connected to the dual curvature measures of convex bodies
introduced by Huang, Lutwak, Yang, and Zhang [27], enriching the variety of
topics that this functional-geometric approach embraces. Also, see [18].

For $K\in\mathcal{K}^{n}$, it is possible to consider more general
perturbations than the one obtained via Minkowski addition, preserving the
structure of (2): These perturbations are obtained as Wulff shapes (also known
as Aleksandrov bodies), which we present in detail in Section 3. The Wulff
shape construction, which depends on a function $f\in C(\mathbb{S}^{n-1})$,
acts on the support function $h_{K}$, and for every $t\in\mathbb{R}$ generates
a (possibly empty) compact convex set $F_{t}K$ whose support function is the
maximal one with the property

$h_{F_{t}K}\leq h_{K}+tf.$

For $t\geq 0$ and $f=h_{L}$ for some $L\in\mathcal{K}^{n}$, $F_{t}K=K+tL$. A
classical lemma due to Aleksandrov (see, for example, [52, Lemma 7.5.3])
generalizes (2) to the formula

$\left.\frac{d}{dt}\mathrm{Vol}(F_{t}K)\right|_{t=0}=\int_{\mathbb{S}^{n-1}}f\,dS_{K}.$
(6)

A clear advantage of (6) over (2) is that it provides a double-sided
derivative This property makes (6) a perfect tool to solve variational
problems, such as the classical Minkowski problem (see, for example, [52,
Section 8]) and its many extensions. The original Minkowski problem can be
formulated as follows: For a given Borel measure $\mathcal{S}$ on
$\mathbb{S}^{n-1}$, find sufficient and necessary conditions so that there
exists $K\in\mathcal{K}^{n}$ such that $\mathcal{S}=S_{K}$. See, for example,
[8, 9, 22, 23, 27, 28, 30, 38, 41, 44] for some recent developments and
advances.

In [17, 49], functional versions of this problem were considered and partially
solved for the measure $\mathcal{M}_{u}$ in (5): For a given Borel measure
$\mathcal{M}$ on $\mathbb{R}^{n}$, find sufficient and necessary conditions so
that there exists $u\in\mathrm{Conv}(\mathbb{R}^{n})$ such that
$\mathcal{M}=\mathcal{M}_{u}$. Further generalizations were explored in [26,
18]. One of the fundamental steps in these works was finding a functional
version of the Wulff shape: For a continuous function
$\zeta:\mathbb{R}^{n}\to\mathbb{R}$ with compact support and
$u\in\mathrm{Conv}(\mathbb{R}^{n}),t\in\mathbb{R}$, many of the previous
authors considered the perturbation

$(u^{*}+t\zeta)^{*},$ (7)

where for a real valued function $g$ the function $g^{*}$ is its Fenchel-
Legendre transform

$g^{*}(x)\coloneqq\sup_{y\in\mathbb{R}^{n}}\\{x\cdot y-u(y)\\}.$

A functional version of (6) can be found in [17, (42)] (later extended in [26,
Lemma 5.2] to more general measures). In [49], lighter hypotheses were
considered on $\zeta$, which was required to be bounded and continuous. In all
these cases, the first variation under the perturbation (7) reads as

$\left.\frac{d}{dt}\mu_{n}((u^{*}+t\zeta)^{*})\right|_{t=0}=\int_{\mathrm{dom}(u)}\zeta(\nabla
u(x))e^{-u(x)}\,dx=\int_{\mathbb{R}^{n}}\zeta(y)\,d\mathcal{M}_{u}(y),$ (8)

where $u\in\mathrm{Conv}(\mathbb{R}^{n})$ is coercive, that is,
$\lim_{|x|\to\infty}u(x)=+\infty$.

A striking difference between the correspondences (2)-(5) and (6)-(8), is that
(8) lacks the boundary term appearing in (5). This is where our work starts:
We introduce a perturbation for convex functions, which we call Wulff-
perturbation, which allows us to obtain a first variation of the type in (8)
where a boundary term appears. We now sketch the construction behind this
perturbation, postponing the full discussion to Section 3. Consider the family
of functions

$C_{rec}(\mathbb{R}^{n})\coloneq\\{\zeta\in
C(\mathbb{R}^{n}):\rho_{\zeta}\text{ exists and is finite and continuous}\\},$

where $\rho_{\zeta}$ is the recession function of $\zeta$, defined as

$\rho_{\zeta}(\boldsymbol{\nu})\coloneqq\lim_{t\to+\infty}\frac{\zeta(t\boldsymbol{\nu})}{t},\quad\boldsymbol{\nu}\in\mathbb{S}^{n-1}.$

To $\zeta\in C_{rec}(\mathbb{R}^{n})$ we associate a certain function
$\overline{\zeta}$ on $\mathbb{S}^{n}$ and then consider its Wulff shape. We
call the convex function representing a specific portion of the boundary of
this Wulff shape the functional Wulff shape of $\zeta$. We restrict our
treatment to the family of convex functions with compact domain,

$\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})\coloneqq\\{u\in\mathrm{Conv}(\mathbb{R}^{n}):\mathrm{dom}(u)\text{
is compact}\\}.$

Then, for a fixed $\zeta\in C_{rec}(\mathbb{R}^{n})$,
$u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, and $t\in\mathbb{R}$, we
consider the functional Wulff shape of $u^{*}+t\zeta$. This is precisely the
Wulff-perturbation, which we denote by $\mathcal{F}_{t}u$. The immediate
application of one of our core results (Theorem 3.14) shows that

$\displaystyle\left.\frac{d}{dt}\mu_{n}(\mathcal{F}_{t}u)\right|_{t=0}=$
$\displaystyle\int_{\mathrm{dom}(u)}\zeta(\nabla
u(x))e^{-u(x)}\,dx+\int_{\mathrm{dom}(u)}\rho_{\zeta}(N_{\mathrm{dom}(u)}(y))e^{-u(x)}\,d\mathcal{H}^{n-1}(y)$
(9) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}\zeta(y)\,d\mathcal{M}_{u}(y)+\int_{\mathbb{S}^{n-1}}\rho_{\zeta}(\boldsymbol{\nu})\,d\mathcal{S}_{u}(\boldsymbol{\nu}).$

Besides the variational formula itself, one of the main improvements is that
our method provides a comprehensive approach to studying all perturbations
mentioned above for functions in
$\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$. Indeed, as we show in Section
5, the expressions (7) and $\mathcal{F}_{t}u$ agree in many instances. In
particular, when $\zeta\in C_{rec}(\mathbb{R}^{n})$ has compact support, or
when $u^{*}+t\zeta$ is convex. The latter hypothesis is always satisfied if
$\zeta=v^{*},v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ and $t\geq 0$
since (see, for example, [48, Theorem 16.4])

$(u^{*}+tv^{*})^{*}=u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v).$

Moreover, $\rho_{v^{*}}=h_{\mathrm{dom}(v)}$ for
$v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, which by (9) provides
immediately (4) in the case
$u,v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$. Choosing $u$ as the
indicator function of a set $K\in\mathcal{K}^{n}$ and $\zeta$ as the
homogeneous extension of a function $f\in C(\mathbb{S}^{n-1})$, one recovers
(6). Still, it is not always guaranteed that the two perturbations coincide!
Details and examples are provided in Section 5. In addition, the results in
this paper suggest that a variational approach can be pursued for inverse
problems connected to the measure $\mathcal{S}_{u}$ appearing in (5).

Our methods allow us, in fact, to prove formulas as (9) for a whole class of
functionals on $\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ similar to (1).
This is achieved through some recent results from the weighted Brunn-Minkowski
theory. See Section 3 for more details and references.

As a byproduct, we can extend (9) in the context of the singular integrals
considered in [26]. In particular, we can remove some hypotheses in the
setting of coercive convex functions, which were needed in [26], and obtain
further improvements in the restricted frame of
$\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ for [26, Theorem 1.3]. In
particular, for $q>0$ we obtain a formula for the one-sided derivative of the
functional

$\mu_{q}(u)\coloneqq\int_{\mathbb{R}^{n}}e^{-u(x)}|x|^{q-n}\,dx,\quad
u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n}).$

######  Theorem 1.2.

Let $u,v\in\mathrm{Conv}(\mathbb{R}^{n})$ and choose $q>0$. If
$0<\mu_{q}(u)<+\infty$, $v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, and
$v(0),u(0)<+\infty$, then,

$\displaystyle\left.\frac{d}{dt}\mu_{q}(u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v))\right|_{t=0^{+}}=$
$\displaystyle\int_{\mathbb{R}^{n}}v^{*}(\nabla u(x))e^{-u(x)}|x|^{q-n}\,dx+$
$\displaystyle\int_{\partial\mathrm{dom}(u)}h_{\mathrm{dom}(v)}(N_{\mathrm{dom}(u)}(y))e^{-u(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y).$

This work is structured as follows. In Section 2, we present some
preliminaries on convex sets and convex functions. In Section 3, we introduce
our notion of functional Wulff shape and, after some technical results on
weighted surface area measures, we prove our main result in Theorem 3.14, from
which (9) will follow. In Section 4, we specialize our techniques to obtain
Theorem 1.2. Finally, in Section 5 we explore the connections between our
notion of Wulff shape and the Fenchel-Legendre transform.

##  2 Preliminaries

The ambient space where we work is the Euclidean space
$\mathbb{R}^{n},n\in\mathbb{N}$. At times, we switch the point of view to
$\mathbb{R}^{n+1}$, and for the standard basis $\\{e_{1},\ldots,e_{n+1}\\}$ we
identify $\mathbb{R}^{n}$ with the hyperplane $H$ orthogonal to $e_{n+1}$. The
origin is denoted by $o$ in both cases. For $x,y\in\mathbb{R}^{n}$ we denote
their scalar product by $x\cdot y$ and write $|x|=\sqrt{x\cdot x}$. The
dimension of the ambient space will always be clear from the context, and we
use the uppercase $X,Y$ when working in $\mathbb{R}^{n+1}$. Given a set
$A\subset\mathbb{R}^{n}$, $\mathrm{int}A$ denotes its interior.

###  2.1 Convex sets

We start by summarizing some notions and results from the classical theory of
convex sets. Of particular interest is the family of non-empty compact convex
sets of $\mathbb{R}^{n}$, denoted by $\mathcal{K}^{n}$, endowed with the
Hausdorff metric. The family of convex bodies, i.e., the elements of
$\mathcal{K}^{n}$ with non-empty interior, is denoted $\mathcal{K}^{n}_{n}$.
For an exhaustive exposition on the topic, see, for example, Schneider’s
monograph [52].

To every convex sets $K$ we can associate a function to $\mathbb{R}^{n}$,
called support function, defined as

$h_{K}(x)\coloneq\sup\\{x\cdot y:y\in K\\},\,x\in\mathbb{R}^{n}.$

Clearly, $h_{K}$ is finite if and only if $K\in\mathcal{K}^{n}$. The behavior
of these functions with respect to Minkowski addition is encoded in the
following property: for every convex set $K,L\subset\mathbb{R}^{n}$

$h_{K}+h_{L}=h_{K+L}.$

A further representation of a convex set $K\subset\mathbb{R}^{n}$ is given by
its indicator function, defined as

$\displaystyle I_{K}(x)\coloneq\begin{cases}0\qquad&\text{ if }x\in K,\\\
+\infty\quad&\text{ otherwise}.\end{cases}$

Another class of functions representing a family of convex sets is the one of
radial functions. If $K\in\mathcal{K}^{n}$ is such that $o\in\mathrm{int}K$,
the radial function of $K$ is defined as

$r_{K}(x)\coloneqq\max\\{s>0:sx\in K\\},\,x\in\mathbb{R}^{n}\setminus\\{o\\}.$

Let $K\in\mathcal{K}^{n}$ and consider its boundary $\partial K$. When $K$ is
a $C^{2}_{+}$ body, the Gauss map

$N_{K}:\partial K\to\mathbb{S}^{n-1}$

gives for each $x\in\partial K$ the unit normal vector to $\partial K$ at $x$,
and is well defined and bijective. In general, for every
$K\in\mathcal{K}^{n}$, the map $N_{K}$ is defined $\mathcal{H}^{n-1}$-almost
everywhere on $\partial K$. In the points $x$ where this is not single-valued,
we consider $N_{K}(x)$ as the unit normal cone of $K$ at $x$, or generalized
Gauss map. This is the set of all the unit vectors $\boldsymbol{\nu}$ such
that $K$ has a tangent hyperplane at $x$ with outer unit normal
$\boldsymbol{\nu}$. Whenever $N_{K}(\boldsymbol{\nu})$ is well defined at
$r_{K}(\boldsymbol{\nu})\boldsymbol{\nu}\in\partial K$, we write
$\alpha_{K}(\boldsymbol{\nu})$ for
$N_{K}(r_{K}(\boldsymbol{\nu})\boldsymbol{\nu})$. By [27, Lemma 2.2], if a
sequence of sets $K_{m}\in\mathcal{K}^{n}$ converges to $K\in\mathcal{K}^{n}$
with respect to the Hausdorff metric, then $\alpha_{K_{m}}$ converges to
$\alpha_{K}$ almost everywhere on $\mathbb{S}^{n-1}$ with respect to the
spherical Lebesgue measure.

The surface area measure of $K$ is defined as

$S_{K}(B)\coloneqq\mathcal{H}^{n-1}(N_{K}^{-1}(B))$ (10)

for every Borel set $B\subset\mathbb{S}^{n-1}$, where $\mathcal{H}^{n-1}$ is
the $(n-1)$-dimensional Hausdorff measure. Surface area measures are finite
Borel measures on $\mathbb{S}^{n-1}$, and they are weakly continuous (see [52,
Section 4.2]). A sequence $(\mu_{m})$ of Borel measures on $\mathbb{S}^{n-1}$
converges weakly to a Borel measure $\mu$ if, for every $f\in
C(\mathbb{S}^{n-1})$,

$\int_{\mathbb{S}^{n-1}}f\,d\mu_{m}\to\int_{\mathbb{S}^{n-1}}f\,d\mu.$

###  2.2 Convex functions

For a function $u:\mathbb{R}^{n}\to\mathbb{R}\cup\\{+\infty\\}$, its epigraph
is the set

$\mathrm{epi}(u)\coloneqq\\{(x,z)\in\mathbb{R}^{n+1}:u(x)\leq z\\}.$

We say that $u$ is a convex function if $\mathrm{epi}(u)$ is a convex set. We
work in the space

$\mathrm{Conv}(\mathbb{R}^{n})\coloneq\\{u:\mathbb{R}^{n}\to\mathbb{R}\cup\\{+\infty\\}\colon
u\text{ is convex, lower semi-continuous, }u\not\equiv+\infty\\}.$

The domain of a convex function is the set

$\mathrm{dom}(u)\coloneq\\{x\in\mathbb{R}^{n}:u(x)<+\infty\\}.$

On the space $\mathrm{Conv}(\mathbb{R}^{n})$, we consider the topology induced
by epi-convergence, characterized as follows: A sequence of functions
$u_{j}\in\mathrm{Conv}(\mathbb{R}^{n})$ epi-converges to
$u\in\mathrm{Conv}(\mathbb{R}^{n})$ if for every $x\in\mathbb{R}^{n}$, the
following conditions are satisfied.

  * •

For every sequence of points $x_{j}\in\mathbb{R}^{n}$ converging to
$x\in\mathbb{R}^{n}$, $u(x)\leq\liminf_{j\to\infty}u_{j}(x_{j})$.

  * •

There exists a sequence $(x_{j})$ converging to $x$, such that
$u(x)=\lim_{j\to\infty}u_{j}(x_{j})$.

Consider the subfamily

$\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R}^{n})\coloneq\\{u\in\mathrm{Conv}(\mathbb{R}^{n}):\lim_{|x|\to\infty}u(x)=+\infty\\}$

of convex coercive functions. Here, convergence of level sets gives a more
intuitive characterization of epi-convergence. For
$u\in\mathrm{Conv}(\mathbb{R}^{n}),t\in\mathbb{R}\cup\\{+\infty\\}$ consider
the sublevel set

$\\{u\leq t\\}\coloneq\\{x\in\mathbb{R}^{n}:u(x)\leq t\\}.$

For a sequence of functions $u_{j}\in\mathrm{Conv}(\mathbb{R}^{n})$ we use the
convention $\\{u_{j}\leq t\\}\to\emptyset$ if there exists
$j_{0}\in\mathbb{N}$ such that $\\{u_{j}\leq t\\}=\emptyset$ for every $j\geq
j_{0}$.

######  Lemma 2.1 ([11, Lemma 5]).

A sequence of functions
$(u_{j})\subset\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R}^{n})$ epi-converges to
$u\in\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R}^{n})$ if and only if
$\\{u_{k}\leq t\\}\to\\{u\leq t\\}$ in the Hausdorff metric for every
$t\in\mathbb{R}$ with $t\neq\min_{x\in\mathbb{R}^{n}}u(x)$.

For $u\in\mathrm{Conv}(\mathbb{R}^{n})$, its Fenchel-Legendre transform is the
function

$u^{*}(x)\coloneq\sup_{y\in\mathbb{R}^{n}}\\{x\cdot y-u(y)\\}.$

Note that since it is the supremum of affine (and thus convex) functions,
$u^{*}$ is again a convex function. Moreover, for a non-convex function
$f:\mathbb{R}^{n}\to\mathbb{R}$, it still makes sense to define $f^{*}$ as
above if we admit the trivial case $f^{*}\equiv+\infty$. In particular,
$f^{*}$ is convex. Moreover, when $u\in\mathrm{Conv}(\mathbb{R}^{n})$ one has
the important fact

$(u^{*})^{*}=u.$

In $\mathrm{Conv}(\mathbb{R}^{n})$, we consider the operations of pointwise
addition and infimal convolution. For $u,v\in\mathrm{Conv}(\mathbb{R}^{n})$
the latter is defined as

$u\square v(x)\coloneq\inf_{y\in\mathbb{R}^{n}}\\{u(y)+v(x-y)\\}, $

and $u\square v\in\mathrm{Conv}(\mathbb{R}^{n})$ if at least one of the two
functions is coercive (which will always be our case). For
$u\in\mathrm{Conv}(\mathbb{R}^{n}),t\geq 0$, instead of the dilation, we have
the two following notions of multiplication: The classical scalar
multiplication and the epi-multiplication

$(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}u)(x)\coloneq
tu\left(\frac{x}{t}\right)$

for $t>0$,
$0\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}u=I_{\\{0\\}}$. Moreover,
one has the fundamental relation

$((t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}u)\square(s\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v))^{*}=tu^{*}+sv^{*}$
(11)

for $u,v\in\mathrm{Conv}(\mathbb{R}^{n})$ and $s,t\geq 0$. Notice that for
$u,v\in\mathrm{Conv}(\mathbb{R}^{n})$ and $t>0$, $\mathrm{epi}(u\square v)$ is
the Minkowski sum $\mathrm{epi}(u)+\mathrm{epi}(v)$, while
$\mathrm{epi}(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)=t(\mathrm{epi}(v))$.

A further meaningful correspondence is the one between support functions and
indicator functions of compact convex sets. Indeed, if $K\in\mathcal{K}^{n}$
and $K$ contains the origin, a quick calculation shows that
$(I_{K})^{*}=h_{K}$.

###  2.3 The space $\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$

In our investigation, a fundamental role is played by the space

$\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})\coloneqq\\{u\in\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R}^{n}):\mathrm{dom}(u)\text{
is compact}\\},$

which is a subset of $\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R}^{n})$. Both
these spaces are endowed with the topology induced by epi-convergence
introduced earlier. The results and notions exposed in the remainder of this
section are from the author and Knoerr [37], where the family
$\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ was introduced as a tool to
infer geometric properties of convex functions through local properties of
compact convex sets.

These functions can be obtained from compact convex sets in $\mathbb{R}^{n+1}$
using the following construction where we use the identification
$H\equiv\mathbb{R}^{n}=e^{\perp}_{n+1}$: To every $K\in\mathcal{K}^{n+1}$ we
associate the function $\lfloor
K\rfloor:\mathbb{R}^{n}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ defined by

$\lfloor K\rfloor(x)\coloneqq\inf\\{t\in\mathbb{R}:(x,t)\in K\\}.$ (12)

In addition, $\lfloor K\rfloor(x)=+\infty$ if and only if $(x,t)\notin K$ for
all $t\in\mathbb{R}$. Analogously, for every $x\in\mathbb{R}^{n}$ and
$K\in\mathcal{K}^{n+1}$ we can define the concave function

$\lceil K\rceil(x)\coloneqq\sup\\{t\in\mathbb{R}:(x,t)\in K\\}.$

In this case $\lceil K\rceil(x)=-\infty$ if $(x,t)\notin K$ for all
$t\in\mathbb{R}$. The following results are proved only for the map
$\lfloor\cdot\rfloor$ for the sake of brevity, but they also hold (up to the
expected adjustments) for the map $\lceil\cdot\rceil$ since for
$K\in\mathcal{K}^{n}$ one has $\lceil K\rceil(x)=-\lfloor R_{H}K\rfloor(x)+c$,
where $R_{H}:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ is the reflection with
respect to $H$ and $c$ is a suitable constant. The following properties can be
found, for example, in [37].

######  Lemma 2.2.

For every $K\in\mathcal{K}^{n+1}$, $\lfloor
K\rfloor\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$. Moreover, the map
$\lfloor\cdot\rfloor:\mathcal{K}^{n+1}\rightarrow\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$
is continuous.

Conversely, we may associate to any
$u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ a convex set in the
following way: We set $M_{u}\coloneqq\max_{x\in\mathrm{dom}(u)}u(x)$, which is
finite since the domain of $u$ is compact and $u$ is convex, and define

$K^{u}\coloneqq\mathrm{epi}(u-M_{u})\cap
R_{H}(\mathrm{epi}(u-M_{u}))+M_{u}e_{n+1},$ (13)

where $R_{H}$ is the reflection with respect to $H$. This is a non-empty
compact and convex set, i.e., $K^{u}\in\mathcal{K}^{n+1}$. We thus obtain a
map

$\begin{gathered}\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})\to\mathcal{K}^{n+1}\\\
\qquad\qquad u\mapsto K^{u}.\end{gathered}$ (14)

By construction $u=\lfloor K^{u}\rfloor$ for
$u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$. This approach resembles the
one adopted in [32], which in turn stems from [4].

Consider the lower half-sphere
$\mathbb{S}^{n}_{-}\coloneqq\\{\boldsymbol{\nu}\in\mathbb{S}^{n}:\boldsymbol{\nu}\cdot
e_{n+1}<0\\}$. We define the lower boundary of $K$ by

$\displaystyle\partial K_{-}\coloneqq\\{X\in\partial
K:N_{K}(X)\cap\mathbb{S}^{n}_{-}\neq\emptyset\\}.$

Notice that the graph of $\lfloor K\rfloor$ coincides with the closure of
$\partial K_{-}$. In particular, if
$u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, then the closure of
$\partial K^{u}_{-}\subset\mathbb{R}^{n+1}$ is the graph of $u$, and we can
parameterize it via the map

$\displaystyle f_{u}:\mathrm{dom}(u)$ $\displaystyle\to\mathbb{R}^{n+1}$
$\displaystyle x$ $\displaystyle\mapsto(x,u(x)).$

If $\gamma:\mathbb{R}^{n+1}\rightarrow\mathbb{R}$ is bounded and Borel
measurable, then by the area formula [42, Theorem 8.1], we obtain

$\int_{\partial{K^{u}_{-}}}\gamma(X)\,d\mathcal{H}(X)=\int_{\mathrm{dom}(u)}\gamma\left((x,u(x))\right)\sqrt{1+|\nabla
u(x)|^{2}}\,dx,$ (15)

where $\sqrt{1+|\nabla u(x)|^{2}}$ is the approximate Jacobian of $f_{u}$.

If $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ is differentiable at
$x\in\mathrm{dom}(u)$, then $\frac{(\nabla u(x),-1)}{\sqrt{1+|\nabla
u(x)|^{2}}}$ is the unique unit outer normal vector to $\mathrm{epi}(u)$ at
$(x,u(x))$. Since $u$ is convex, it is differentiable almost everywhere, and
thus, the unit normal vectors to the epigraph are defined almost everywhere.
We have the following.

######  Lemma 2.3 ([37, Corollary 3.7]).

For every $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ and $\eta\in
C(\mathbb{S}_{-}^{n})$

$\displaystyle\int_{\mathbb{S}_{-}^{n}}\eta(\boldsymbol{\nu})\,dS_{K^{u}}(\boldsymbol{\nu})=\int_{\mathrm{dom}(u)}\eta\left(\frac{(\nabla
u(x),-1)}{\sqrt{1+|\nabla u(x)|^{2}}}\right)\sqrt{1+|\nabla u(x)|^{2}}\,dx.$
(16)

We conclude this section by recalling the parallelism between support
functions and the Fenchel-Legendre transform. As we will see in Section 5,
this parallelism morally works for every function in
$\mathrm{Conv}(\mathbb{R}^{n})$. Consider $K\in\mathcal{K}^{n+1}$. Via
explicit calculations, we infer

$\displaystyle h_{K}(y,-1)=$ $\displaystyle\sup\\{X\cdot(y,-1)\colon X\in
K\\}=\sup\\{X\cdot(y,-1)\colon X\in\partial K\\}$ (17) $\displaystyle=$
$\displaystyle\sup\\{(x,\lfloor K\rfloor(x))\cdot(y,-1)\colon
x\in\mathrm{dom}(\lfloor K\rfloor)\\}$ $\displaystyle=$
$\displaystyle\sup\\{x\cdot y-\lfloor K\rfloor(x)\colon
x\in\mathrm{dom}(\lfloor K\rfloor)\\}$ $\displaystyle=$
$\displaystyle\sup\\{x\cdot y-\lfloor K\rfloor(x)\colon
x\in\mathbb{R}^{n}\\}={\lfloor K\rfloor}^{*}(y).$

When $K=K^{u}$ for some $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$,
(17) takes the form

$h_{K^{u}}(y,-1)=u^{*}(y)$ (18)

The map

$K\mapsto h_{K}(\cdot,-1)$

was already considered, for example, by Alesker [1] and by Knoerr [34] to
create a correspondence between $\mathcal{K}^{n+1}$ and the space of finite
convex functions. Equation (17) shows that the latter point of view and the
one presented here are dual.

##  3 First variations for measures of epigraphs

Consider on $\mathbb{R}^{n+1}$ a measure $\mu$ such that

$d\mu(x,z)=d\eta(x)d\omega(z),x\in\mathbb{R}^{n}\equiv H,z\in\mathbb{R}\equiv
H^{\perp},$

where $\eta$ and $\omega$ are positive Borel measures on $\mathbb{R}^{n}$ and
$\mathbb{R}$ respectively. The space of interest is always some subset of the
family of functions $\mathrm{Conv}(\mathbb{R}^{n})$, and we want to evaluate
the measure of the epigraph of $u$ through $\mu$, that is (using the abuse of
notation $\mu(u)\coloneqq\mu(\mathrm{epi}(u))$)

$\mu(u)=\int_{\mathrm{epi}(u)}\,d\mu(z,x)=\int_{\mathrm{dom}(u)}\int_{u(x)}^{+\infty}\,d\omega(z)d\eta(x).$

Ignoring, for now, the various summability assumptions, if we define
$\Phi(t)=\omega([t,+\infty))$ we can write

$\mu(u)=\int_{\mathrm{dom}(u)}\Phi(u(x))\,d\eta(x).$

We focus on the case where there exist $\phi\in C(\mathbb{R})$ and $\psi\in
C(\mathbb{R}^{n})$ such that $d\omega(z)=\phi(z)dz$ and $d\eta(x)=\psi(x)dx$.
We now introduce a suitable deformation process for functions in
$\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, which we build upon the
classical notion of Wulff shape.

###  3.1 Wulff shapes of convex functions

The concept of Wulff shape, introduced more than a century ago by Wulff [55],
is nowadays a well-established scientific tool, especially in the study of the
shapes of crystals. Significant developments have been obtained throughout the
20th century from the mathematical perspective. See, for example, the work of
Fonseca [19].

######  Definition 3.1.

Consider a function $f:\mathbb{S}^{n-1}\to\mathbb{R}$. Its Wulff shape is the
set

$[f]\coloneq\bigcap_{\boldsymbol{\nu}\in\mathbb{S}^{n-1}}H^{-}_{\boldsymbol{\nu}}(f(\boldsymbol{\nu})),$

where $H^{-}_{\boldsymbol{\nu}}(t)\coloneq\\{x\in\
\mathbb{R}^{n}\colon\boldsymbol{\nu}\cdot x\leq t\\}$ is the negative closed
half-space with outer normal $\boldsymbol{\nu}$ and distance $t$ from the
origin. Equivalently, $[f]$ is the unique maximal (with respect to inclusion)
convex set satisfying the condition

$h_{[f]}(\boldsymbol{\nu})\leq f(\boldsymbol{\nu})\text{ for every
}\boldsymbol{\nu}\in\mathbb{S}^{n-1}.$ (19)

######  Remark 3.2.

Notice that if $f>c>0$, $[f]$ is clearly non-empty. In general, if
$\ell_{y}(x)\coloneq y\cdot x,y\in\mathbb{R}^{n}$ and $f-\ell_{y}>c>0$, then
$[f]$ is non-empty and $y$ is in the interior of $[f]$. In particular, if
$[f]$ is non-empty,

$[f+\ell_{y}]=[f]+y,$ (20)

for every $y\in\mathbb{R}^{n}$ (notice that $\ell_{y}=h_{\\{y\\}}$). Indeed,
by (19), for every $y\in\mathbb{R}^{n}$

$h_{[f]+y}=h_{[f]}+\ell_{y}\leq f+\ell_{y}.$

If, by contradiction, $h_{[f]+y}$ was not maximal for $f+\ell_{y}$, neither
would be $h_{[f]}$ for $f$, which would contradict (19), proving (20).

We now propose an extension of the same construction to convex functions in
$\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$. To the best of our knowledge,
this is the first time that this idea has been implemented in this fashion.
However, many of the techniques we present are scattered around the literature
and deeply encode the process at hand. See, for example, [17, 50]. We work
with $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ and $\zeta\in
C(\mathbb{R}^{n})$ such that its recession function

$\rho_{\zeta}(\boldsymbol{\nu})\coloneqq\lim_{t\to+\infty}\frac{\zeta(t\boldsymbol{\nu})}{t},\quad\boldsymbol{\nu}\in\mathbb{S}^{n-1}$

exists, is finite, and depends continuously on $\boldsymbol{\nu}$, and define
the family of functions

$C_{rec}(\mathbb{R}^{n})\coloneq\\{\zeta\in
C(\mathbb{R}^{n}):\rho_{\zeta}\text{ exists and is finite and continuous}\\}.$

For $\zeta\in C_{rec}(\mathbb{R}^{n})$ we consider the function $\bar{\zeta}$
on $\mathbb{S}^{n}_{-}$ obtained as

$\bar{\zeta}(\boldsymbol{\nu})\coloneq\frac{\zeta\left(g(\boldsymbol{\nu})\right)}{\sqrt{1+\left|g(\boldsymbol{\nu})\right|^{2}}}.$
(21)

Here $g:\mathbb{S}^{n}_{-}\to\mathbb{R}^{n}$ is the gnomonic projection

$\boldsymbol{\nu}\mapsto\frac{\boldsymbol{\nu}-(e_{n+1}\cdot\boldsymbol{\nu})e_{n+1}}{e_{n+1}\cdot\boldsymbol{\nu}}.$
(22)

We will make use of the extension of $\bar{\zeta}$ on the whole
$\mathbb{S}^{n}$ obtained by reflection on $H$, that is, if
$\boldsymbol{\nu}=(\nu_{1},\dots,\nu_{n+1})\in\mathbb{S}^{n}_{-}$,
$\bar{\zeta}(\boldsymbol{\nu})=\bar{\zeta}((\nu_{1},\dots,\nu_{n},-\nu_{n+1}))$.
When $\nu_{n+1}=0$, $\bar{\zeta}$ is extended by continuity. This extension
exists and is finite since $\zeta\in C_{rec}(\mathbb{R}^{n})$. Note that with
the identification $\mathbb{S}^{n}\cap H\equiv\mathbb{S}^{n-1}$ the continuous
extension of the function $\bar{\zeta}$ to $\mathbb{S}^{n-1}$ is equal to
$\rho_{\zeta}$. To make the notation lighter, we refer to $\bar{\zeta}$ both
for the transform (21) and the extension. We now present the functional
counterpart of Definition 3.1. We remark that the Wulff shapes we are now
considering are in $\mathbb{R}^{n+1}$.

######  Definition 3.3.

For $\zeta\in C_{rec}(\mathbb{R}^{n})$ we define the functional Wulff shape of
$\zeta$ as the function

$\lfloor\zeta\rfloor\coloneqq\left\lfloor\left[\,\overline{\zeta}\,\right]\right\rfloor$

whenever $\left[\,\overline{\zeta}\,\right]$ is non-empty. If
$\left[\,\overline{\zeta}\,\right]$ is empty, we set
$\lfloor\zeta\rfloor\neq+\infty$.

######  Remark 3.4.

For $v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ such that
$o\in\mathrm{int}(\mathrm{epi}(v))$ and $\max_{y\in\mathrm{dom}(v)}v(y)=0$,
the choice $\zeta=v^{*}$ reduces to the particular case $\lfloor
v^{*}\rfloor=v$. In general, though, this is a misleading intuition, and in
Section 5, we show that for a non-convex $\zeta\in C_{rec}(\mathbb{R}^{n})$
its functional Wulff shape and Fenchel-Legendre transform provide two
different functions.

Following Remark 3.2, notice that if
$v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ we can always find
$(x,z)\in\mathbb{R}^{n+1}$ such that
$o\in\mathrm{int}(\mathrm{epi}(v\square(I_{\\{x\\}}+z)))$ and
$\max_{y\in\mathrm{dom}(v)}v(y-x)+z=0$. Thus, up to a suitable translation of
$\mathrm{epi}(v)$, the construction in Definition 3.3 agrees with the Fenchel-
Legendre transform for functions in
$\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$. We will make use of this fact
in the proof of Theorem 1.2.

For $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, consider now the
associated compact convex set $K^{u}$ associated as in (13), and $\zeta\in
C_{rec}(\mathbb{R}^{n})$. Then we have on $\mathbb{S}^{n}$ the function

$h_{u,t}(\boldsymbol{\nu})\coloneq
h_{K^{u}}(\boldsymbol{\nu})+t\bar{\zeta}(\boldsymbol{\nu}).$ (23)

Consider its classical Wulff shape $[h_{u,t}]$, which we denote by $K^{u,t}$.
The function $h_{u,t}$ can be extended to $\mathbb{R}^{n+1}$ considering its
positively homogeneous extension $h_{u,t}(X)=|X|h_{u,t}(X/|X|)$ for
$X\in\mathbb{R}^{n+1},|X|\neq 0$. We now specialize Definition 3.3,
introducing the Wulff-perturbation of $u$ with respect to $\zeta$ at time $t$.
The dependence on $\zeta$ is omitted for a lighter notation since it will
always be clear from the context.

######  Definition 3.5.

Consider $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n}),\zeta\in
C_{rec}(\mathbb{R}^{n})$, and $t\in\mathbb{R}$. Then the function

$\mathcal{F}_{t}u\coloneqq\lfloor K^{u,t}\rfloor$

where $K^{u,t}$ is obtained by (23), is the Wulff-perturbation of $u$ with
respect to $\zeta$ at time $t$.

In the rest of the paper, we use both $\mathcal{F}_{t}u$ and $\lfloor
K^{u,t}\rfloor$ depending on the convenience of the notation and the point of
view we want to highlight (geometric or functional). The following fact will
be fundamental in the proof of our main result.

######  Lemma 3.6.

Let $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, and $\zeta\in
C_{rec}(\mathbb{R}^{n})$. Consider the segment $\ell_{\tau}=\\{se_{n+1}\colon
s\in[0,\tau]\\},\tau>0$. Then

$[h_{K^{u}+\ell_{\tau}}+t\bar{\zeta}]=[h_{K^{u}}+t\bar{\zeta}]+\ell_{\tau}.$

###### Proof.

Consider $K_{\tau,t}\coloneq[h_{K^{u}+\ell_{\tau}}+t\bar{\zeta}]$. Notice that
$h_{\ell_{\tau}}(\boldsymbol{\nu})=0$ for every
$\boldsymbol{\nu}\in\mathbb{S}^{n}$ such that $N\cdot e_{n+1}\leq 0$ and
therefore in these directions

$h_{K^{u}+\ell_{\tau}}(\boldsymbol{\nu})+t\bar{\zeta}(\boldsymbol{\nu})=h_{K^{u}}(\boldsymbol{\nu})+h_{\ell_{\tau}}(\boldsymbol{\nu})+t\bar{\zeta}(\boldsymbol{\nu})=h_{K^{u}}(\boldsymbol{\nu})+t\bar{\zeta}(\boldsymbol{\nu}).$

In particular, this is sufficient to grant that

$\lfloor
K_{\tau,t}\rfloor=\lfloor[h_{K^{u}+\ell_{\tau}}+t\bar{\zeta}]\rfloor=\lfloor[h_{K^{u}}+t\bar{\zeta}]\rfloor=\lfloor
K^{u,t}\rfloor.$

Notice that, by construction, $K^{u,t}$ is symmetric with respect to
$e_{n+1}^{\perp}$ up to a suitable vertical translation. With the notation
$\tilde{\mathcal{F}}_{t}u\coloneqq\lceil K^{u,t}\rceil$, this implies that
$\mathcal{F}_{t}u$ and $-\tilde{\mathcal{F}}_{t}u$ are equal up to a constant.
Now, if $e_{n+1}\cdot N\geq 0$, we have

$\displaystyle
h_{K^{u}+\ell_{\tau}}(\boldsymbol{\nu})+t\bar{\zeta}(\boldsymbol{\nu})=h_{K^{u}}(\boldsymbol{\nu})+h_{\ell_{\tau}}(\boldsymbol{\nu})+t\bar{\zeta}(\boldsymbol{\nu})=$
$\displaystyle h_{K^{u}}(\boldsymbol{\nu})+\tau e_{n+1}\cdot
N+t\bar{\zeta}(\boldsymbol{\nu})=h_{K^{u}}(\boldsymbol{\nu})+h_{\\{\tau
e_{n+1}\\}}(\boldsymbol{\nu})+t\bar{\zeta}(\boldsymbol{\nu}).$

By (20),

$[h_{K^{u}}+h_{\\{\tau e_{n+1}\\}}+t\bar{\zeta}]=[h_{K^{u}}+t\bar{\zeta}]+\tau
e_{n+1}=K^{u,t}+\tau e_{n+1}.$

Therefore,

$\displaystyle\lceil[h_{K^{u}+\ell_{\tau}}+t\bar{\zeta}]\rceil=\lceil[h_{K^{u}}+h_{\\{\tau
e_{n+1}\\}}+t\bar{\zeta}]\rceil=\lceil[h_{K^{u}}+t\bar{\zeta}]\rceil+\tau=\tilde{\mathcal{F}}_{t}u+\tau.$

By definition,

$\displaystyle K_{\tau,t}=$ $\displaystyle\left(\bigcap_{\boldsymbol{\nu}\cdot
e_{n+1}\leq
0}H^{-}_{\boldsymbol{\nu}}(h_{K^{u}+\ell_{\tau}}(\boldsymbol{\nu})+t\bar{\zeta}(\boldsymbol{\nu}))\right)$
$\displaystyle\cap$ $\displaystyle\left(\bigcap_{\boldsymbol{\nu}\cdot
e_{n+1}\geq
0}H^{-}_{\boldsymbol{\nu}}(h_{K^{u}+\ell_{\tau}}(\boldsymbol{\nu})+t\bar{\zeta}(\boldsymbol{\nu}))\right)$
$\displaystyle=$
$\displaystyle\mathrm{epi}\left(\mathcal{F}_{t}u\right)\cap\mathrm{epi}\left(\tilde{\mathcal{F}}_{t}u+\tau\right)=K^{u,t}+\ell_{\tau},$

concluding the proof. ∎

The core ideas for the proof of our main result are encoded in the following
properties of Wulff shapes proved by Willson [54]. Consider
$K\in\mathcal{K}^{n+1},t\geq 0$ and $f\in C(\mathbb{S}^{n})$; we use the
notation $F_{t}K$ for the Wulff shape of $h_{K}+tf$, that is

$F_{t}K\coloneqq[h_{K}+tf],$ (24)

The dependence on a function $f$ will always be clear from the context.
Theorems 5.1 and 5.6 from [54] read as follows.

######  Theorem 3.7.

If $K_{m}\to K$ in $\mathcal{K}^{n+1}$, $t_{m}\to t_{0}$ in $\mathbb{R}$ and
$F_{t_{0}}K$ has non-empty interior, then $F_{t_{m}}K_{m}$ has non-empty
interior for $m$ large and $F_{t_{m}}K_{m}\to F_{t_{0}}K$ in
$\mathcal{K}^{n+1}$.

In particular, Theorem 3.7 implies that $F_{t}K$ is continuous in $t$.

######  Theorem 3.8.

Let $s$ and $t$ be nonnegative real numbers. Let
$K\in\mathcal{K}^{n+1}_{n+1},f\in C(\mathbb{S}^{n-1})$ and assume $F_{t}K$ has
non-empty interior. Then

$F_{s}F_{t}K=F_{s+t}K$

These properties will be very useful later to obtain differentiability in $t$
for the measure of $F_{t}K$, since by Lemma 2.2, all these properties are
passed on to functional Wulff shapes.

###  3.2 Weighted Brunn-Minkowski theory

Formula (9) is an application of our results, which are built on a wider
framework. Indeed, (1) can be interpreted as the measure $\mu$ on
$\mathbb{R}^{n+1}$ of the epigraph of $u$, where
$d\mu(x,z)=e^{-z}dxdz,x\in\mathbb{R}^{n},z\in\mathbb{R}$. In recent years has
drawn much interest the so-called weighted Brunn-Minkowski theory, where one
considers instead of $\mathrm{Vol}$ a generic measure $\mu$ with continuous
density $\Psi$ with respect to the Lebesgue measure. See, for example,
Livshyts [41], Alonso-Gutierrez, Hernández Cifre, Roysdon, Yepes Nicolàs, and
Zvavitch [2], Kryvonos and Langharst [38], and Fradelizi, Langharst, Madiman
and Zvavitch [20]. We note that the mentioned papers concern compact convex
sets, while recently Schneider [53] explored these ideas in the context of
pseudocones, which, in our language, are epigraphs of functions in a suitable
subfamily of $\mathrm{Conv}(\mathbb{R}^{n})$.

We consider the following generalized notion of surface area measure in (10),
that is, the weighted surface area measure, which is defined as

$S_{\mu,K}(B)=\int_{N_{K}^{-1}(B)}\Psi(X)\,d\mathcal{H}^{n}(X)$

for every $K\in\mathcal{K}^{n}$ and Borel set $B\subset\mathbb{S}^{n}$. A core
ingredient of our approach is Lemma 2.7 from [38], which generalizes
Aleksandrov’s variational lemma [52, Lemma 7.5.3].

######  Lemma 3.9 (Kryvonos and Langharst).

Let $\mu$ be a Borel measure on $\mathbb{R}^{n+1}$ with continuous density
$\Psi$ with respect to the Lebesgue measure. Then for $f\in C(\mathbb{S}^{n})$
and $K\in\mathcal{K}^{n+1}_{n+1}$, we have

$\left.\frac{d}{dt}\mu([h_{K}+tf])\right|_{t=0}=\int_{\mathbb{S}^{n}}f(\boldsymbol{\nu})\,dS_{\mu,K}(\boldsymbol{\nu})=\int_{\partial
K}f(N_{K}(X))\Psi(X)\,d\mathcal{H}^{n}(X).$ (25)

######  Remark 3.10.

Lemma 3.9 was originally formulated with the additional assumption
$o\in\mathrm{int}K$, which was unnecessary since the boundary structure of a
compact convex set is invariant under translations. Thus, both the set and the
measure can be suitably translated so that (25) still holds. For completeness,
and since it will be a useful remark later, we provide a proof of this fact.
Suppose that the origin is not contained in $K$. For every point $Y$ in the
interior of $K$, $h_{K}-\ell_{Y}>0$, where $\ell_{Y}(X)=Y\cdot
X,X\in\mathbb{R}^{n+1}$, and $K-Y$ has the origin in its interior. Then, by
(20), we can consider the Wulff shape $[h_{K}-\ell_{Y}+tf]$ to infer by Lemma
3.9

$\left.\frac{d}{dt}\mu([h_{K}-\ell_{Y}+tf])\right|_{t=0}=\int_{\partial(K-Y)}f(N_{K-Y}(X))\Psi(X)\,d\mathcal{H}^{n}(X).$

Consider $\tilde{\Psi}(\cdot)=\Psi(\cdot+Y)$ and the measure $\tilde{\mu}$
which has $\tilde{\Psi}$ as density. By Lemma 3.9, the translation invariance
of the Hausdorff measure, and since $N_{K-Y}(X)=N_{K}(X+Y)$,

$\displaystyle\left.\frac{d}{dt}\mu([h_{K}+tf])\right|_{t=0}=$
$\displaystyle\left.\frac{d}{dt}\tilde{\mu}([h_{K}-\ell_{Y}+tf])\right|_{t=0^{+}}=$
$\displaystyle\int_{\partial(K-Y)}f(N_{K-Y}(X))\tilde{\Psi}(X)\,d\mathcal{H}^{n}(X)=$
$\displaystyle\int_{\partial
K}f(N_{K}(Z))\tilde{\Psi}(Z-Y)\,d\mathcal{H}^{n}(Z)$ $\displaystyle=$
$\displaystyle\int_{\partial K}f(N_{K}(Z))\Psi(Z)\,d\mathcal{H}^{n}(Z),$

proving that in Lemma 3.9, we do not need the origin to belong to the interior
of $K$.

We now show that the weighted volume is continuous for compact convex sets
with non-empty interior, and the weighted surface area measure is weakly
continuous. The latter fact follows implicitly, for continuous densities, from
the stability result by Livschyts [41, Proposition A.3]. For our applications,
we include the possibility of working with singular densities. Our ambient
space is still $\mathbb{R}^{n+1}$ with the formalism introduced before, where
we write $X\in\mathbb{R}^{n+1}$ as $X=(x,z),x\in\mathbb{R}^{n}\equiv
H,z\in\mathbb{R}\equiv\mathrm{span}\\{e_{n+1}\\}$.

######  Lemma 3.11.

For $q>0$ and $\Psi\in C(\mathbb{R}^{n+1})$ non-negative, consider the measure
$\mu$ on $\mathbb{R}^{n+1}$ with density $d\mu(X)=\Psi(X)|x|^{q-n}dxdz$, and a
sequence $(K_{m})\subset\mathcal{K}^{n+1}$ converging to
$K\in\mathcal{K}^{n+1}_{n+1}$ in the Hausdorff metric. In case $q<n$, suppose
moreover that
$o\in\mathrm{pr}_{H}\left(\mathrm{int}K\cap\bigcap_{m\in\mathbb{N}}\mathrm{int}K_{m}\right)$,
where $\mathrm{pr}_{H}$ is the orthogonal projection on
$H\equiv\mathbb{R}^{n}$. Then

  * •

The volume is continuous (this conclusion is trivial for $q\geq n$), that is,

$\mu(K_{m})\to\mu(K),$

  * •

The sequence of measures $S_{\mu,K_{m}}$ converges weakly to $S_{\mu,K}$.

###### Proof.

First, notice that without loss of generality, we can suppose
$o\in\left(\mathrm{int}K\cap\bigcap_{m\geq M}\mathrm{int}K_{m}\right)$ for $M$
sufficiently large. Indeed, if $q\geq n$, we can follow Remark 3.10. If
$0<q<n$, thanks to the additional hypothesis
$o\in\mathrm{pr}_{H}\left(\mathrm{int}K\cap\bigcap_{m\in\mathbb{N}}\mathrm{int}K_{m}\right)$
we just need a vertical translation, which will affect only $\Psi$ proceeding
as in the same remark.

Under this assumption, there exists $\delta>1$ such that

$\frac{1}{\delta}B^{n}\subset K_{m}\subset\delta B^{n}$

for every $m$ sufficiently large. This chain of inclusions is equivalent to

$\frac{1}{\delta}<r_{K_{m}}<\delta$ (26)

for $m$ every sufficiently large, and since $K_{m}\to K$ in the Hausdorff
metric the sequence of radial functions $r_{K_{m}}$ converges uniformly to
$r_{K}$. Moreover, for $f\in C(\mathbb{S}^{n})$ we have that
$\alpha_{K_{m}}(\boldsymbol{\nu})\to\alpha_{K}(\boldsymbol{\nu})$ for almost
every $\boldsymbol{\nu}\in\mathbb{S}^{n}$ with respect to the spherical
Lebesgue measure.

By direct computations using polar coordinates,

$\displaystyle\mu(K_{m})=$
$\displaystyle\frac{1}{n}\int_{\mathbb{S}^{n}}\int_{0}^{r_{K_{m}}(\boldsymbol{\nu})}\psi(t\boldsymbol{\nu})|\mathrm{pr}_{H}(t\boldsymbol{\nu})|^{q-n}t^{n}\,dtd\boldsymbol{\nu}$
$\displaystyle=$
$\displaystyle\frac{1}{n}\int_{\mathbb{S}^{n}}|\mathrm{pr}_{H}(\boldsymbol{\nu})|^{q-n}\left(\int_{0}^{r_{K_{m}}(\boldsymbol{\nu})}\psi(t\boldsymbol{\nu})t^{q}\,dt\right)d\boldsymbol{\nu}.$

The radial functions $r_{K_{m}}$ are continuous and converge uniformly to
$r_{K}$. Therefore

$\int_{0}^{r_{K_{m}}(\boldsymbol{\nu})}\psi(t\boldsymbol{\nu})t^{q}\,dt\to\int_{0}^{r_{K}(\boldsymbol{\nu})}\psi(t\boldsymbol{\nu})t^{q}\,dt$

uniformly, and the continuity of $\mu$ is proved once we show that

$\int_{\mathbb{S}^{n}}|\mathrm{pr}_{H}(\boldsymbol{\nu})|^{q-n}\,d\boldsymbol{\nu}<+\infty.$
(27)

Consider indeed the representation

$\int_{\mathbb{S}^{n}}|\mathrm{pr}_{H}(\boldsymbol{\nu})|^{q-n}\,d\boldsymbol{\nu}=2\int_{\mathbb{S}^{n}_{-}}|\mathrm{pr}_{H}(\boldsymbol{\nu})|^{q-n}\,d\boldsymbol{\nu}.$

By (15)

$\int_{\mathbb{S}^{n}_{-}}|\mathrm{pr}_{H}(\boldsymbol{\nu})|^{q-n}\,d\boldsymbol{\nu}=\int_{B^{n}}\frac{|x|^{q-n+1}}{\sqrt{1-|x|^{2}}}\,dx=\int_{\mathbb{S}^{n-1}}\int_{0}^{1}\frac{t^{q}}{\sqrt{1-t^{2}}}\,dtd\boldsymbol{\nu},$

which is finite by direct computations. Thus, we have (27).

For the second statement, we adapt the argument in [28, Theorem 3.4]. By the
definition of weighted surface area measure and [27, Lemma 2.9], we infer

$\displaystyle\int_{\mathbb{S}^{n}}f(\boldsymbol{\nu})\,dS_{\mu,K_{m}}(\boldsymbol{\nu})=\int_{\partial
K_{m}}f(N_{K}(X))\Psi(X)|x|^{q-n}\,d\mathcal{H}^{n}((x,z))$
$\displaystyle=\int_{\mathbb{S}^{n}}\frac{f(\alpha_{K_{m}}(\boldsymbol{\nu}))}{\boldsymbol{\nu}\cdot\alpha_{K_{m}}(\boldsymbol{\nu})}\Psi(r_{K_{m}}(\boldsymbol{\nu})\boldsymbol{\nu})|\mathrm{pr}_{H}(r_{K_{m}}(\boldsymbol{\nu})\boldsymbol{\nu})|^{q-n}r_{K_{m}}^{n}(\boldsymbol{\nu})\,d\boldsymbol{\nu}.$

By (26) and since $f$ and $\Psi$ are continuous, there exists $C>0$ such that

$\frac{f(\alpha_{K_{m}}(\boldsymbol{\nu}))}{\boldsymbol{\nu}\cdot\alpha_{K_{m}}(\boldsymbol{\nu})}\Psi(r_{K_{m}}(\boldsymbol{\nu})\boldsymbol{\nu})|\mathrm{pr}_{H}(r_{K_{m}}(\boldsymbol{\nu})\boldsymbol{\nu})|^{q-n}r_{K_{m}}^{n}(\boldsymbol{\nu})\leq\delta^{n+1+(q-n)\mathrm{sign}(q-n)}C|\mathrm{pr}_{H}(\boldsymbol{\nu})|^{q-n}$

for every $m$ and for almost every $\boldsymbol{\nu}\in\mathbb{S}^{n}$, where
$\mathrm{sign}(\tau)=\tau/|\tau|$ if $0\neq\tau\in\mathbb{R}$ and
$\mathrm{sign}(0)=0$. Notice that the expression on the left converges almost
everywhere to

$\frac{f(\alpha_{K}(\boldsymbol{\nu}))}{\boldsymbol{\nu}\cdot\alpha_{K}(\boldsymbol{\nu})}\Psi(r_{K}(\boldsymbol{\nu})\boldsymbol{\nu})|\mathrm{pr}_{H}(r_{K}(\boldsymbol{\nu})\boldsymbol{\nu})|^{q-n}r_{K}^{n}(\boldsymbol{\nu}).$

By (27) and a direct application of the dominated convergence theorem, the
proof is concluded. ∎

Notice that the estimates provided in the proof imply that for every
$K\in\mathcal{K}^{n+1}$ such that $o\in\mathrm{pr}_{H}(\mathrm{int}K)$ the
values $\mu(K)$ and $S_{\mu,K}(\mathbb{S}^{n})$ are finite.

We now apply Lemma 3.11 to show that Lemma 3.9 holds for the measures we are
considering. For the convenience of the reader, we recall the following
classical result (see, for example, [51, Theorem 7.17]).

######  Lemma 3.12.

Suppose $f_{m}:[a,b]\to\mathbb{R},m\in\mathbb{N}$ is a sequence of functions,
differentiable on $[a,b]$ and such that $f_{m}(x_{0})$ converges for some
$x_{0}\in[a,b]$. If the derivatives $f^{\prime}_{m}$ converge uniformly on
$[a,b]$, then $f_{m}$ converges uniformly on $[a,b]$, to a function $f$, and

$f^{\prime}(x)=\lim_{m\to\infty}f^{\prime}_{m}(x)$

for every $x\in[a,b]$.

We treat only the case $0<q<n$, since for $q\geq n$ the following is a direct
consequence of Lemma 3.9.

######  Corollary 3.13.

Let $\mu$ be a Borel measure on $\mathbb{R}^{n+1}$ with density
$d\mu(X)=\Psi(X)|x|^{q-n}dxdz$, $0<q<n$. Then for $f\in C(\mathbb{S}^{n})$ and
$K\in\mathcal{K}^{n+1}_{n+1}$ such that $o\in\mathrm{pr}_{H}(\mathrm{int}K)$,
we have

$\left.\frac{d}{dt}\mu([h_{K}+tf])\right|_{t=0}=\int_{\mathbb{S}^{n}}f(\boldsymbol{\nu})\,dS_{\mu,K}(\boldsymbol{\nu}).$

###### Proof.

Again, without loss of generality, we can suppose that $o\in\mathrm{int}K$.
Consider the sequence of measures $\mu_{m}$ defined by
$d\mu_{m}(x,z)=\Psi(X)(|x|\wedge m)^{q-n}$, where $a\wedge
b\coloneqq\min\\{a,b\\}$ for $a,b\in\mathbb{R}$. Then, for every $m$, the
density of $\mu_{m}(x,z)$ with respect to the Lebesgue measure is a continuous
function on $\mathbb{R}^{n+1}$. Moreover, since $o\in\mathrm{int}K$, by
Theorem 3.7 there exist $\varepsilon>0$ such that $o\in\mathrm{int}[F_{t}K]$
for every $t\in[0,\varepsilon]$, where $F_{t}K=[h_{K}+tf]$. In particular,
$\mu_{m}(K_{t})$ is continuous in $[0,\varepsilon]$ for every $m$.

By Theorem 3.8, for every $t_{0}\in[0,\varepsilon)$ and $t>0$ sufficiently
small

$\frac{\mu_{m}(F_{t_{0}+t}K)-\mu_{m}(F_{t}K)}{t}=\frac{\mu_{m}(F_{t}F_{t_{0}}K)-\mu_{m}(F_{t_{0}}K)}{t}.$

Therefore, Lemma 3.9 shows that $\mu_{m}(F_{t}K)$ has continuous derivative on
$[0,\varepsilon]$. Then since $\mu_{m}(K_{t})$ is continuous, it is
differentiable in $[0,\varepsilon]$ for every $m$.

Consider the decomposition $f=f^{+}-f^{-}$, where $f^{+}=(f\vee 0)$ and
$f^{-}=-(f\wedge 0)$. Then

$\int_{\mathbb{S}^{n}}f^{+}(\boldsymbol{\nu})\,dS_{\mu_{m},K}(\boldsymbol{\nu})\to\int_{\mathbb{S}^{n}}f^{+}(\boldsymbol{\nu})\,dS_{\mu,K}(\boldsymbol{\nu})$

increasingly and pointwise in $t\in[0,\varepsilon]$. The same holds if we
replace $f^{+}$ with $f^{-}$. By Lemma 3.11 the two limits are continuous and
bounded, and therefore

$\int_{\mathbb{S}^{n}}f(\boldsymbol{\nu})\,dS_{\mu_{m},K}(\boldsymbol{\nu})\to\int_{\mathbb{S}^{n}}f(\boldsymbol{\nu})\,dS_{\mu,K}(\boldsymbol{\nu})$

uniformly in $t$ (see, for example, [51, Theorem 7.13]).

The proof is concluded since we satisfy all the hypotheses of Lemma 3.12 and
thus

$\lim_{m\to\infty}\lim_{t\to
0}\frac{\mu_{m}([h_{K}+tf])-\mu_{m}(K)}{t}=\lim_{t\to
0}\lim_{m\to\infty}\frac{\mu_{m}([h_{K}+tf])-\mu_{m}(K)}{t}.$

∎

###  3.3 Proof of the variational formula

We are finally ready to prove our core result. It reads as follows.

######  Theorem 3.14.

Let $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ such that
$\mathrm{int}(\mathrm{dom}(u))\neq\emptyset$, and $\zeta\in
C_{rec}(\mathbb{R}^{n})$. Consider, moreover, a measure $\mu$ on
$\mathbb{R}^{n+1}$ such that $d\mu(z,x)=\phi(z)\psi(x)dz\,dx$ with non-
negative functions $\phi\in C(\mathbb{R})\cap L^{1}([a,+\infty))$ and $\psi\in
C(\mathbb{R}^{n})$ for some $a\in\mathbb{R}$ such that $\phi(z)\to 0$ as
$z\to+\infty$. Then

$\displaystyle\left.\frac{d}{dt}\mu(\mathcal{F}_{t}u)\right|_{t=0}$
$\displaystyle=$ $\displaystyle\int_{\mathrm{dom}(u)}\zeta(\nabla
u(x))\phi(u(x))\psi(x)\,dx+\int_{\partial\mathrm{dom}(u)}$
$\displaystyle\rho_{\zeta}(N_{\mathrm{dom}(u)}(y))\Phi(u(y))\psi(x)\,d\mathcal{H}^{n-1}(y),$

exists and is finite, where $\Phi(t)=\int_{t}^{+\infty}\phi(z)dz$.

Notice that, under these assumptions, the measure of the epigraph of $u$ is
finite for every $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$.

As anticipated, our strategy is to work between convex sets and convex
functions. To perform this passage formally, we introduce the family

$\displaystyle\mathcal{K}^{n+1}_{+}\coloneq$
$\displaystyle\\{K\subset\mathbb{R}^{n+1}:K\text{ is convex, closed, with
nonempty interior, and }\text{pr}_{H}(K)\in\mathcal{K}^{n}_{n}\\}.$

Notice that this family includes precisely $\mathcal{K}^{n+1}_{n+1}$ and the
epigraphs of the functions in $\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$
with $n$-dimensional domain, where $\mathbb{R}^{n}\equiv H=e_{n+1}^{\perp}$.
Let $K,L\in\mathcal{K}^{n+1}_{+}$ and let $\mu$ be a measure on
$\mathbb{R}^{n+1}$. The $\mu$-symmetric-difference between $K$ and $L$ is

$d_{\mu}(K,L)\coloneq\mu(K\Delta L)=\int_{\mathbb{R}^{n+1}}\chi_{K\Delta
L}(x)\,d\mu(x),$

where for a measurable set $A$ we denote by $\chi_{A}$ is characteristic
function. In particular, if we use the Gaussian measure $\gamma_{n+1}$ on
$\mathbb{R}^{n+1}$, defined by its density

$d\gamma_{n+1}(x)=\frac{1}{\sqrt{2\pi}^{n+1}}e^{-|x|^{2}/2}\,dx,$

then $d_{\gamma_{n+1}}$ defines a metric on $\mathcal{K}^{n+1}_{+}$. This
follows from the convexity of the involved sets and the sets being of
dimension $n+1$. Notice that $d_{\gamma_{n+1}}(K,L)\in[0,1]$ for every
$K,L\in\mathcal{K}^{n+1}_{+}$. This kind of metric was studied in a wider
generality by Li and Mussnig [40]. Here we provide a proof that
$d_{\gamma_{n+1}}$ induces an appropriate metric on $\mathcal{K}^{n+1}_{+}$,
that is, we can approximate functions in
$\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ by convex bodies in
$\mathcal{K}^{n+1}_{n+1}$.

######  Lemma 3.15.

Let $\gamma_{n+1}$ be the Gaussian measure on $\mathbb{R}^{n+1}$. The function
$d_{\gamma_{n+1}}:\mathcal{K}^{n+1}_{+}\times\mathcal{K}^{n+1}_{+}\to[0,1]$ is
a distance. Moreover, its restrictions to $\mathcal{K}^{n+1}_{n+1}$ and the
family of epigraphs of functions in
$\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ induce the topology of the
Hausdorff metric and epi-convergence, respectively.

###### Proof.

The equivalence of epi-convergence and convergence with respect to
$d_{\gamma_{n+1}}$ can be proved analogously to Theorem 1.2 in [40].

To prove that $d_{\gamma_{n+1}}$ induces the Hausdorff metric on
$\mathcal{K}^{n+1}_{n+1}$, we use the first part of the proof as follows: by
Lemma 2.1, the Hausdorff convergence of a sequence
$\mathcal{K}_{m}\in\mathcal{K}^{n+1}_{n+1}$ to some
$\mathcal{K}\in\mathcal{K}^{n+1}_{n+1}$ is equivalent to the epi-convergence
of $I_{K_{m}}$ to $I_{K}$ as functions on $\mathbb{R}^{n+1}$. However, as we
just proved, this is equivalent to the convergence of
$\mathrm{epi}(I_{K_{m}})$ to $\mathrm{epi}(I_{K})$ with respect to
$d_{\gamma_{n+2}}$. Direct calculations show that

$d_{\gamma_{n+2}}(\mathrm{epi}(I_{K_{m}}),\mathrm{epi}(I_{K}))=Cd_{\gamma_{n+1}}(K_{m},K)$

for some absolute constant $C>0$, concluding the proof. ∎

With these instruments at hand, we can start the proof of Theorem 3.14.

###### Proof of Theorem 3.14.

Let us first sketch the outline of the proof. We start by appropriately re-
writing the variational formula (25) with $K=K^{v}$ for some
$v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, $\Psi(x,z)=\phi(z)\psi(x)$,
and $f=\bar{\zeta}$ as defined in (21). For $\tilde{v}=\lceil K^{v}\rceil$,
direct computations using (15) show

$\begin{split}&\lim_{t\to
0}\frac{\mu([h_{K^{v}}+t\bar{\zeta}])-\mu(K^{v})}{t}=\int_{\partial
K^{v}}\bar{\zeta}(N_{K^{v}}((x,z)))\phi(z)\psi(x)\,d\mathcal{H}^{n}((x,z))\\\
&=\int_{\mathrm{dom}(v)}\zeta(\nabla v(x))\phi(v(x))\psi(x)\,dx\\\
&+\int_{\mathrm{dom}(v)}\zeta(\nabla v(x))\phi(\tilde{v}(x))\psi(x)\,dx\\\
&+\int_{\partial\mathrm{dom}(v)}\bar{\zeta}((N_{\mathrm{dom}(v)}(x),0))\left(\int_{v(x)}^{\tilde{v}(x)}\phi(s)\,ds\right)\psi(x)\,d\mathcal{H}^{n-1}(x).\end{split}$
(28)

In particular, notice that $\bar{\zeta}$ restricted to the equator is by
definition $\rho_{\zeta}$. We have used the following fact: Since $v$ and
$\tilde{v}$ are one a suitable reflection of the other, $\nabla
v(x)=-\nabla\tilde{v}(x)$ for every $x$ where the gradient exists. For $v$,
the normal to the epigraph expressed through the gradient is $(\nabla
v,-1)/\sqrt{1+|\nabla v|^{2}}$, while for $\tilde{v}$ we have
$(-\nabla\tilde{v},1)/\sqrt{1+|\nabla\tilde{v}|^{2}}$. Moreover, by
construction
$\bar{\zeta}\left(N_{K^{v}}(x,z)\right)=\bar{\zeta}\left(R_{H}(N_{K^{v}}(x,z))\right)$.
Since

$\bar{\zeta}\left(R_{H}(N_{K^{v}}(x,z))\right)=\bar{\zeta}\left(\frac{-\nabla\tilde{v}(x),1}{\sqrt{1+|\nabla\tilde{v}(x)|^{2}}}\right)=\bar{\zeta}\left(\frac{\nabla
v(x),1}{\sqrt{1+|\nabla v(x)|^{2}}}\right)=\zeta(\nabla v(x)),$

we have the integral on the third line of (28).

From here, for a fixed $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, the
idea is to approximate $\mathrm{epi}(\mathcal{F}_{t}u)$ with a suitable
sequence of convex bodies in order to obtain our claim as a limit of the
integrals of the form (28). Then, we conclude by checking the hypotheses
required to apply Lemma 3.12. The proof is structured in three steps after
introducing the sequence approximating $\mathrm{epi}(\mathcal{F}_{t}u)$.

Clearly, dim$(\mathrm{epi}(u))=n+1$. Consider now the segment
$\ell_{m}=\\{se_{n+1}\colon s\in[0,m]\\}$ for $m\in\mathbb{N}$. Then,
$K^{u}+\ell_{m}\to\mathrm{epi}(u)$ in the symmetric-difference distance
$d_{\gamma_{n+1}}$ as $m\to\infty$. Consider the sequence of Wulff shapes
$K_{m,t}\coloneq F_{t}(K^{u}+\ell_{m})=[h_{K^{u}+\ell_{m}}+t\bar{\zeta}]$.
Notice that $K_{m,0}$ is monotonic with respect to inclusion, and thus the
sequence $\mu(K_{m,0})$ is increasing. By the properties of the sequence
$K_{m,0}$ and the measure $\mu$ it follows from the monotone convergence
theorem that

$\lim_{m\to\infty}\mu(K_{m,0})=\mu(\mathrm{epi}(u_{0}))=\mu(\mathrm{epi}(u)).$

Moreover, notice that the sequence of sets $K_{m,t}$ converges to
$\mathrm{epi}(u_{t})$ in the symmetric-difference distance topology
monotonically. In particular, Lemma 3.6 implies that
$K_{m,t}=K^{u,t}+\ell_{m}$, where
$K^{u,t}=F_{t}(K^{u})=[h_{K^{u}}+t\bar{\zeta}]$.

Step 1(Limit as $t\to 0$). Define now $u_{m}\coloneq\lfloor
K^{u}+\ell_{m}\rfloor$ and $\tilde{u}_{m}\coloneq\lceil K^{u}+\ell_{m}\rceil$.
Notice that $u_{m}=u$ and $\tilde{u}_{m}=\tilde{u}+m$, where $\tilde{u}=\lceil
K^{u}\rceil$. Then, replacing $K^{v}$ with $K^{u}+\ell_{m}$ in (28), we infer

$\begin{split}&\lim_{t\to
0}\frac{\mu(K_{m,t})-\mu(K^{u}+\ell_{m})}{t}=\int_{\partial(K^{u}+\ell_{m})}f(N_{K^{u}+\ell_{m}}((x,z)))\phi(z)\psi(x)\,d\mathcal{H}^{n}((x,z))\\\
&=\int_{\mathrm{dom}(u)}\zeta(\nabla u(x))\phi(u(x))\psi(x)\,dx\\\
&+\int_{\mathrm{dom}(u)}\zeta(\nabla u(x))\phi(\tilde{u}(x)+m)\psi(x)\,dx\\\
&+\int_{\partial\mathrm{dom}(u)}\rho_{\zeta}((N_{\mathrm{dom}(u)}(x))\left(\int_{u(x)}^{\tilde{u}(x)+m}\phi(s)\,ds\right)\psi(x)\,d\mathcal{H}^{n-1}(x).\end{split}$
(29)

Notice that for $m$ fixed, the integrals in the last three lines of (29) are
all finite since they are just a suitable decomposition of the right-hand side
of the first line. We claim that the right-hand side of (29) converges to

$\int_{\mathrm{dom}(u)}\zeta(\nabla
u(x))\phi(u(x))\psi(x)\,dx+\int_{\partial\mathrm{dom}(u)}\rho_{\zeta}(N_{\mathrm{dom}(u)}(y))\Phi(u(y))\psi(x)\,d\mathcal{H}^{n-1}(y)$

as $m\to\infty$. The first integral is determined by the graph of $u$ and thus
is fixed by the sequence we are considering. Concerning the second one, since
$\tilde{u}_{m}=\tilde{u}+m$ is the parametrization of the upper part of
$\partial(K^{u}+\ell_{m})$, we infer
$\lim_{m\to\infty}\tilde{u}_{m}(x)=+\infty$ for every $x\in\mathrm{dom}(u)$.
Therefore $\lim_{m\to\infty}\phi(\tilde{u}_{m}(x))=0$ for every
$x\in\mathrm{dom}(u)$ since $\phi$ converges to $0$ by hypothesis. Then,

$\displaystyle\left|\int_{\mathrm{dom}(u)}\zeta(\nabla
u(x))\phi(\tilde{u}(x)+m)\psi(x)\,dx\right|\leq$
$\displaystyle\max_{x\in\mathrm{dom}(u)}\phi(\tilde{u}(x)+m)\int_{\mathrm{dom}(u)}\left|\zeta(\nabla
u(x))\psi(x)\right|\,dx.$

Since the integral on the right-hand side of the inequality is finite (it is
part of the weighted surface area measure of $K^{u}$ when in Lemma 3.9 we
consider $\Psi(x,z)=\psi(x)$) and

$\max_{x\in\mathrm{dom}(u)}\phi(\tilde{u}(x)+m)\leq\sup_{y\in[\inf\tilde{u}+m,+\infty]}\phi(y)\to
0$

as $m\to\infty$, the integral on the third line of (29) converges to $0$.

We conclude this step observing that

$\int_{\partial\mathrm{dom}(u)}\rho_{\zeta}(N_{\mathrm{dom}(u)}(x))\left(\int_{u(x)}^{\tilde{u}_{m}(x)}\phi(s)\,ds\right)\psi(x)\,d\mathcal{H}^{n-1}(x)$

converges to

$\int_{\partial\mathrm{dom}(u)}\rho_{\zeta}(N_{\mathrm{dom}(u)}(x))\Phi(u(x))\psi(x)\,d\mathcal{H}^{n-1}(x)$

as $m\to\infty$. Indeed $\tilde{u}_{m}(x)\to\infty$ as $m\to\infty$, thus

$\int_{u(x)}^{\tilde{u}_{m}(x)}\phi(s)\,ds\to\int_{u(x)}^{+\infty}\phi(s)\,ds=\Phi(u(x))$

increasingly, and the limit is finite by hypothesis. Then since

$\displaystyle\left|\rho_{\zeta}(N_{\mathrm{dom}(u)}(x))\left(\int_{u(x)}^{\tilde{u}_{m}(x)}\phi(s)\,ds\right)\psi(x)\right|\leq\left|\rho_{\zeta}(N_{\mathrm{dom}(u)}(x))\psi(x)\right|\Phi(u(x)),$

the dominated convergence theorem grants the desired convergence.

Step 2(The derivative exists). We now prove that $\mu(K_{m,t})$ is
differentiable with respect to $t$ for each $t$ in $[0,\varepsilon]$ for every
$m\in\mathbb{N}$ and $\varepsilon$ suitably small, and its derivative
converges uniformly as $m\to\infty$ on $[0,\varepsilon]$ to

$\int_{\mathrm{dom}(u_{t})}\zeta(\nabla\mathcal{F}_{t}u(x))\phi(\mathcal{F}_{t}u(x))\psi(x)\,dx+\int_{\partial\mathrm{dom}(u_{t})}\rho_{\zeta}(N_{\mathrm{dom}(u)}(x))\Phi(\mathcal{F}_{t}u(x))\psi(x)\,d\mathcal{H}^{n-1}(x).$

For $m$ fixed, the differentiability of $\mu(K_{m,t})$ follows from Theorem
3.8. Indeed this theorem implies

$\frac{\mu(F_{t+t_{0}}(K^{u}+\ell_{m}))-\mu(F_{t_{0}}(K^{u}+\ell_{m}))}{t}=\frac{\mu(F_{t}F_{t_{0}}(K^{u}+\ell_{m}))-\mu(F_{t_{0}}(K^{u}+\ell_{m}))}{t},$

and as long as $K_{m,t_{0}}$ has non-empty interior we can apply Lemma 3.9 and
have, for $t_{0}\in[0,\varepsilon]$,

$\frac{d\mu(K_{m,t})}{dt}\Big{|}_{t=t_{0}}=\int_{\mathbb{S}^{n}}\bar{\zeta}(\boldsymbol{\nu})\,dS_{\mu,K_{m,t_{0}}}(\boldsymbol{\nu}).$
(30)

Notice that $K^{u}+\ell_{m}$ has non-empty interior for every $m$ since
$K^{u}+\ell_{1}\subset K^{u}+\ell_{m}$. Thus, we can choose $\varepsilon$ such
that (30) remains true for every $m$ and for every $t_{0}\in[0,\varepsilon]$.
Since $K_{m,t}$ is continuous in $t$ by Theorem 3.7 and $\,dS_{\mu,K_{m,t}}$
is weakly continuous by Lemma 3.11 (in the case $q=n$) , the right derivative
of $\mu(K_{m,t})$ is continuous in $t$ on $[0,\varepsilon]$. Since if the
derivative of a continuous function is continuous, then the function itself is
differentiable, the function $\mu(K_{m,t})$ is differentiable in
$(0,\varepsilon)$ for every $m$, as desired.

Step 3(Uniform convergence). To prove the uniform convergence of the
derivatives (30), we start by repeating the procedure for the limits of (29)
for $t\in[0,\varepsilon]$, $\varepsilon>0$ as chosen in the previous step.
Consider the decomposition in (29) applied to $K_{m,t}$ for a general
$t\in[0,\varepsilon]$. The first integral is independent of $m$. Indeed
$\lfloor K_{m,t}\rfloor=\mathcal{F}_{t}u$ for every $m$ since
$K_{m,t}=K^{u,t}+\ell_{m}$. Furthermore,
$\mathrm{dom}(u_{m,t})=\mathrm{dom}(u_{t})$ for every $m$. For the second
integral, notice that since for $t\to 0$ one has $\lceil
K_{m,t}\rceil\eqcolon\tilde{u}_{m,t}\to\tilde{u}_{m}=\tilde{u}+m$, we can find
a sequence of values $M_{m}\to+\infty$ as $m\to\infty$ such that for every
$t\in[0,\varepsilon]$ we have
$M_{m}\leq\min_{\mathrm{dom}(\mathcal{F}_{t}u)}\tilde{u}_{m,t}$. Moreover,
notice that for every $x\in\mathrm{dom}(\mathcal{F}_{t}u)$ we have
$\nabla\mathcal{F}_{t}u(x)=-\nabla\tilde{u}_{m,t}(x)$ since by Lemma 3.6
$\tilde{u}_{m,t}$ is the reflection of $\mathcal{F}_{t}u$ with respect to $H$
up to a constant. Then, for the second integral in (29), we have

$\displaystyle\left|\int_{\mathrm{dom}(\mathcal{F}_{t}u)}\zeta(\nabla\mathcal{F}_{t}u(x))\phi(\tilde{u}_{m,t}(x))\psi(x)\,dx\right|$
$\displaystyle\leq$
$\displaystyle\left(\sup_{t\in[M_{m},+\infty]}\phi(t)\right)\int_{\mathrm{dom}(\mathcal{F}_{t}u)}|\zeta(\nabla\mathcal{F}_{t}u(x))\psi(x)|\,dx$
$\displaystyle=$
$\displaystyle\left(\sup_{t\in[M_{m},+\infty]}\phi(t)\right)\int_{\mathrm{dom}(\mathcal{F}_{t}u)}|\zeta(\nabla\mathcal{F}_{t}u(x))\psi(x)|\,dx$
$\displaystyle=$
$\displaystyle\left(\sup_{t\in[M_{m},+\infty]}\phi(t)\right)\int_{{\partial
K^{u,t}_{-}}}|\bar{\zeta}(g^{-1}(\nabla\mathcal{F}_{t}u(x)))|\psi(x)\,d\mathcal{H}^{n}((x,z))$
$\displaystyle\leq$
$\displaystyle\left(\sup_{t\in[M_{m},+\infty]}\phi(t)\right)\max_{\boldsymbol{\nu}\in\mathbb{S}^{n}}|\bar{\zeta}(\boldsymbol{\nu})|\max_{t\in[0,\varepsilon]}\left(\int_{\partial
K^{u,t}}\psi(x)\,d\mathcal{H}^{n}((x,z))\right),$

where $g$ was defined in (22), and therefore the first line converges to $0$
independently of $t$. In the last inequality we have used that $\phi\geq 0$
and $\partial K^{u,t}_{-}\subset\partial K^{u,t}$. Notice that the maximum in
$t$ is bounded since $K^{u,t}$ is continuous in $t$. Finally, for the last
integral, the convergence is granted again by the dominated convergence
theorem and making use of the fact that
$\mathrm{dom}(u_{t})=\mathrm{dom}(u_{m,t})$. Indeed, we get

$\begin{split}&\left|\int_{\partial\mathrm{dom}(\mathcal{F}_{t}u)}\rho_{\zeta}(N_{\mathrm{dom}(\mathcal{F}_{t}u)}(x))\Phi(\mathcal{F}_{t}u(x))\psi(x)\,d\mathcal{H}^{n-1}(x)-\right.\\\
&\left.\int_{\partial\mathrm{dom}(u_{m,t})}\rho_{\zeta}(N_{\mathrm{dom}(u_{m,t})}(x))\left(\int_{\mathcal{F}_{t}u(x)}^{\tilde{u}_{m,t}}\phi(s)\,ds\right)\psi(x)\,d\mathcal{H}^{n-1}(x)\right|\\\
=&\left|\int_{\partial\mathrm{dom}(\mathcal{F}_{t}u)}\rho_{\zeta}(N_{\mathrm{dom}(u_{m,t})}(x))\left(\int_{\tilde{u}_{m,t}(x)}^{+\infty}\phi(s)\,ds\right)\psi(x)\,d\mathcal{H}^{n-1}(x)\right|\\\
\leq&\max_{\boldsymbol{\nu}\in\mathbb{S}^{n-1}}|\rho_{\zeta}(\boldsymbol{\nu})|\max_{t\in[0,\varepsilon]}\left|\int_{\partial\mathrm{dom}(\mathcal{F}_{t}u)}\psi(x)\,d\mathcal{H}^{n-1}(x)\right|\left(\int_{C+m/2}^{+\infty}\phi(s)\,ds\right),\end{split}$

where
$C=\max_{t\in[0,\varepsilon]}\max_{x\in\mathrm{dom}(\mathcal{F}_{t}u)}\mathcal{F}_{t}u(x)$.
Notice that the maximum on $[0,\varepsilon]$ is finite since the integral is
continuous in $t$, as $\mathrm{dom}(\mathcal{F}_{t}u)$ is the projection on
$e_{n+1}^{\perp}$ of $K^{u,t}$, which is continuous in $t$. Then, as
$m\to\infty$, the last integral converges to $0$ uniformly on $t$.

Conclusion. Having concluded all the steps, we can now safely apply Lemma 3.12
with $f_{m}(t)=\mu(K_{m,t})$, concluding the proof. ∎

A refinement of the same proof provides the following statement.

######  Theorem 3.16.

Let $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ such that
$o\in\mathrm{int}(\mathrm{dom}(u))$, and $\zeta\in C_{rec}(\mathbb{R}^{n})$.
Consider, moreover, a measure $\mu$ on $\mathbb{R}^{n+1}$ such that
$d\mu(z,x)=\phi(z)\psi(x)|x|^{q-n}dz\,dx,q>0$, with $\phi\in C(\mathbb{R})\cap
L^{1}([a,+\infty))$ for some $a\in\mathbb{R}$ and $\psi\in C(\mathbb{R}^{n})$
such that $\phi(z)\to 0$ as $z\to+\infty$. Then

$\displaystyle\left.\frac{d}{dt}\mu_{n}(\mathcal{F}_{t}u)\right|_{t=0}$
$\displaystyle=$ $\displaystyle\int_{\mathrm{dom}(u)}\zeta(\nabla
u(x))\phi(u(x))\psi(x)|x|^{q-n}\,dx+\int_{\partial\mathrm{dom}(u)}$
$\displaystyle\rho_{\zeta}(N_{\mathrm{dom}(u)}(y))\Phi(u(y))\psi(x)|y|^{q-n}\,d\mathcal{H}^{n-1}(y),,$

exists and is finite, where $\Phi(t)=\int_{t}^{+\infty}\phi(z)dz$.

###### Proof.

Consider the sets $K_{m,t}$ obtained through the same construction as in the
proof of Theorem 3.14. Since $o\in\mathrm{int}(\mathrm{dom}(u))$, for
$\varepsilon>0$ suitably small $o\in\mathrm{pr_{H}}(\mathrm{int}K_{m,t})$ for
every $m$. Then, we can repeat the proof verbatim since Lemma 3.11 and
Corollary 3.13 can be applied again. ∎

##  4 Application to moment measures

Consider the case $\phi(z)=e^{-z}$, $\psi\equiv 1$. This point of view has
been initially investigated by Colesanti and Fragalà [10] and Cordero-
Erausquin and Klartag [17]. The two interpretations stem from independent
perspectives. The former aims to generalize some classical concepts from the
Brunn-Minkowski theory. The latter originates from the interest in the KLS
conjecture and solutions of differential equations in Kähler-Einstein
manifolds; see, for example, Klartag [33]. The first approach required $C^{2}$
regularity on the interior of the domain, while the second was restricted to
essentially continuous convex functions (see [17, Definition 2]). Recently,
Rotem [49, 50] improved significantly these results, dropping all regularity
assumptions with the result in Theorem 1.1. Theorem 3.14 immediately implies
the following variant.

######  Corollary 4.1.

Consider $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, $\zeta\in
C_{rec}(\mathbb{R}^{n})$, and the measure $\mu$ in $\mathbb{R}^{n+1}$ such
that $d\mu(x,z)=e^{-z}dzdx$. Then,

$\left.\frac{d}{dt}\mu_{n}(\mathcal{F}_{t}u)\right|_{t=0}=\int_{\mathrm{dom}(u)}\zeta(\nabla
u(x))e^{-u(x)}\,dx+\int_{\partial\mathrm{dom}(u)}\rho_{\zeta}(N_{\mathrm{dom}(u)}(y))e^{-u(y)}\,d\mathcal{H}^{n-1}(y).$

We can then move on to the measures studied in [26], which, in our setting,
correspond to the choice $d\mu(x,z)=d\mu_{q}(x,z)=e^{-z}|x|^{q-n}dxdz$. The
functional volume then is

$\mu_{q}(u)\coloneqq\int_{\mathrm{dom}(u)}e^{-u(x)}|x|^{q-n}\,dx$

for $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ such that
$o\in\mathrm{int}(\mathrm{dom}(u))$. As proved in [26, Proposition 1.2], if
$u$ is coercive, this integral is finite, which is always the case if
$u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$. Moreover, Theorem 1.3 in
the same work provides the following variational result.

######  Theorem 4.2 (Huang, Liu, Xi, and Zhao).

Consider $u\in\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R}^{n})$ such that
$o\in\mathrm{int}(\mathrm{dom}(u))$, and choose $q>0$. Assume $u$ achieves its
minimum at the origin and

$\limsup_{|x|\to 0}\frac{|e^{-u(x)}-e^{-u(0)}|}{|x|^{\alpha+1}}<\infty,$ (31)

for some $0<\alpha<1$.

Consider, moreover, $v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ such
that $o\in\mathrm{dom}(v)$. Then,

$\displaystyle\lim_{t\to
0^{+}}\frac{\mu_{q}(\mathrm{epi}(u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)))-\mu_{q}(\mathrm{epi}(u))}{t}=$
$\displaystyle\int_{\mathbb{R}^{n}}v^{*}(\nabla u(x))$ $\displaystyle
e^{-u(x)}|x|^{q-n}\,dx+\int_{\partial\mathrm{dom}(u)}h_{\mathrm{dom}(v)}(N_{\mathrm{dom}(u)}(y))e^{-u(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y).$

A direct application of Theorem 3.16 shows that the hypothesis (31) is not
always necessary for $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$.
Moreover, we can consider all deformations in $C_{rec}(\mathbb{R}^{n})$.
Compare with [26, Lemma 5.2].

######  Corollary 4.3.

Consider $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ and $\zeta\in
C_{rec}(\mathbb{R}^{n})$, and $\mu$ the measure in $\mathbb{R}^{n+1}$ such
that $d\mu(x,z)=e^{-z}|x|^{q-n}dzdx$. Suppose, moreover that
$o\in\mathrm{int}(\mathrm{dom}(u))$. Then

$\left.\frac{d}{dt}\mu_{q}(\mathcal{F}_{t}u)\right|_{t=0}=\int_{\mathbb{R}^{n}}\zeta(\nabla
u(x))e^{-u(x)}|x|^{q-n}\,dx+\int_{\partial\mathrm{dom}(u)}\rho_{\zeta}(N_{\mathrm{dom}(u)}(y))e^{-u(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y).$
(32)

The two integrals in (32) are finite whenever
$u\in\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R}^{n})$ and $\zeta$ is Lipschitz
(thus including the case
$\zeta=v^{*},v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$), as the
following lemma shows. The finiteness of the first one is a consequence of
[26, Theorem 5.12], while for the second one, we adapt the idea from [50,
Proposition 1.6], which in turn is inspired by an argument by Ball [6].

######  Lemma 4.4.

Consider $u\in\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R}^{n})$ such that
$o\in\mathrm{int}(\mathrm{dom}(u))$. If $\zeta\in C_{rec}(\mathbb{R}^{n})$ is
Lipschitz, the integrals

$\displaystyle\int_{\mathbb{R}^{n}}\left|\zeta(\nabla
u(x))\right|e^{-u(x)}|x|^{q-n}\,dx\text{ and }$
$\displaystyle\int_{\partial\mathrm{dom}(u)}\left|\rho_{\zeta}(N_{\mathrm{dom}(u)}(y))\right|e^{-u(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)$

are finite.

###### Proof.

First notice that since $\zeta$ is a Lipschitz function there exists $L\geq 0$
such that for every $x\in\mathbb{R}^{n}$

$|\zeta(x)|-|\zeta(0)|\leq|\zeta|-|\zeta(0)||\leq L|x|.$

That is, there exists $a\geq 0$ such that

$|\zeta(x)|\leq L|x|+a$

for every $x\in\mathbb{R}^{n}$. Then

$\int_{\mathbb{R}^{n}}\left|\zeta(\nabla
u(x))\right|e^{-u(x)}|x|^{q-n}\,dx\leq\int_{\mathbb{R}^{n}}L|\nabla
u(x)|e^{-u(x)}|x|^{q-n}\,dx+\int_{\mathbb{R}^{n}}ae^{-u(x)}|x|^{q-n}\,dx.$

The first integral on the right is finite thanks to [26, Theorem 5.12], while
the second is finite by [26, Proposition 1.2] since $u$ is coercive.

Consider now the second integral in the statement. Since $u$ is convex and
coercive there exist $A,c>0$ such that $e^{-u(x)}\leq Ae^{-c|x|}$ (see [31,
Lemma 2.1]). Moreover, notice that

$e^{-c|x|}=c\int_{0}^{+\infty}e^{-ct}\mathds{1}_{tB^{n}}(x)dt$

for every $x\in\mathbb{R}^{n}$, where $\mathds{1}_{tB^{n}}=e^{-I_{tB^{n}}}$ is
the characteristic function of the Euclidean unit ball. Then we infer

$\begin{split}\int_{\partial\mathrm{dom}(u)}e^{-u(y)}&|y|^{q-n}\,d\mathcal{H}^{n-1}(y)\leq
A\int_{\partial\mathrm{dom}(u)}e^{-c|y|}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)\\\
=Ac&\int_{\partial\mathrm{dom}(u)}\int_{0}^{+\infty}e^{-ct}\mathds{1}_{tB^{n}}(y)|y|^{q-n}\,dtd\mathcal{H}^{n-1}(y)\\\
=Ac&\int_{0}^{+\infty}e^{-ct}\int_{\partial\mathrm{dom}(u)\cap
tB^{n}}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)dt.\end{split}$ (33)

If $0<q<n$ we continue (33) by

$\displaystyle\ldots\leq Ac\left(\int_{0}^{1}e^{-ct}\int_{\partial
tB^{n}}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)\,dt+\int_{1}^{+\infty}e^{-ct}\int_{\partial
tB^{n}}\,d\mathcal{H}^{n-1}(y)\,dt\right)$
$\displaystyle=Ac\mathcal{H}^{n-1}(\mathbb{S}^{n-1})\left(\int_{0}^{1}e^{-ct}t^{q-1}\,dt+\int_{1}^{+\infty}e^{-ct}t^{n-1}\,dt\right).$

If, alternatively, $n\leq q$, analogously we obtain

$\displaystyle\ldots\leq Ac\left(\int_{0}^{1}e^{-ct}\int_{\partial
tB^{n}}\,d\mathcal{H}^{n-1}(y)\,dt+\int_{1}^{+\infty}e^{-ct}\int_{\partial
tB^{n}}\,d\mathcal{H}^{n-1}(y)|y|^{q-n}\,dt\right)$
$\displaystyle=Ac\mathcal{H}^{n-1}(\mathbb{S}^{n-1})\left(\int_{0}^{1}e^{-ct}t^{n-1}\,dt+\int_{1}^{+\infty}e^{-ct}t^{q-1}\,dt\right).$

In any case, since the integrals in both the last lines are finite, all the
preceding ones are. Observing that

$\int_{\partial\mathrm{dom}(u)}\left|\rho_{\zeta}(N_{\mathrm{dom}(u)}(y))\right|e^{-u(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)\leq
L\int_{\partial\mathrm{dom}(u)}e^{-u(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y),$

the proof is concluded. ∎

We now report two estimates from the proof of [26, Theorem 5.12]. We use again
the fact that if $u\in\mathrm{Conv}(\mathbb{R}^{n})$ is coercive then there
exist $A,c>0$ such that $e^{-u(x)}\leq Ae^{c|x|}$. In this lemma we write
$(x_{1},\dots,x_{n})=x\in\mathbb{R}^{n}$ as
$x=(\bar{x},x_{n}),\bar{x}\in\mathbb{R}^{n-1}$.

######  Lemma 4.5.

Consider $u\in\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R}^{n})$ and let $A,c>0$ be
such that $e^{-u(x)}\leq Ae^{-c|x|}$ for every $x\in\mathbb{R}^{n}$. Then, for
every $m\in\mathbb{N}$:

  * •

If $0<q\leq n$

$\begin{split}&\int_{\mathbb{R}^{n}\setminus
mB^{n}}\left|\frac{\partial}{\partial
x_{n}}e^{-u(x)}\right||x|^{q-n}\,dx\leq\\\
4A&\int_{\left(\mathbb{R}^{n}\setminus\frac{m}{2}B^{n}\right)\cap\mathbb{R}^{n-1}}\sup_{x_{n}\in\mathbb{R}}e^{-c|(\bar{x},x_{n})|}|\bar{x}|^{q-n}\,d\bar{x}\\\
+4A&\left(\frac{\sqrt{3}}{2}m\right)^{q-n}\int_{\frac{m}{2}B^{n}\cap\mathbb{R}^{n-1}}\sup_{x_{n}\in\mathbb{R}}e^{-c|(\bar{x},x_{n})|}\,d\bar{x},\end{split}$
(34)

  * •

If $q>n$ there exists a constant $C$ (independent of $u$)

$\begin{split}&\int_{\mathbb{R}^{n}\setminus
mB^{n}}\left|\frac{\partial}{\partial
x_{n}}e^{-u(x)}\right||x|^{q-n}\,dx\leq\\\
4CA\sum_{k=m}^{\infty}&(k+1)^{q-n}\left((k+1)^{n-1}e^{-c(k-1)}+(k-1)^{n-1}e^{c\sqrt{k}}\right).\end{split}$
(35)

Furthermore, we need the following approximation result.

######  Lemma 4.6.

Consider $u\in\mathrm{Conv}(\mathbb{R}^{n})$,
$v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ and $t\geq 0$. Then, for
every $m>\max_{y\in\mathrm{dom}(v)}|y|$ there exists $\delta(m)>0$ such that
$\lim_{m\to\infty}\delta(m)=+\infty$ and

$u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)(x)=(u+I_{m^{\prime}B^{n}})\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)(x)$

for every $|x|\leq\delta(m),0\leq t\leq 1$, and $m^{\prime}\geq m$.

###### Proof.

We choose $\delta(m)\coloneq m-\max_{y\in\mathrm{dom}(v)}|y|$. Notice that, by
construction,

$u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)(x)\leq(u+I_{mB^{n}})\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)(x)$

for every $x\in\mathbb{R}^{n}$ and every $m$.

Consider $m>\max_{y\in\mathrm{dom}(v)}|y|$ and $|x|\leq\delta(m)$. If
$u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)(x)=+\infty$,
there is nothing to prove. If
$u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)(x)<+\infty$,
then necessarily $x\subset\mathrm{dom}(v)+mB^{n}$. Therefore,

$u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)(x)=\inf_{y\in
mB^{n}}\\{u(y)+t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v(x-y)\\}=\inf_{y\in\mathbb{R}^{n}}\\{u(y)+I_{mB^{n}}(y)+t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v(x-y)\\}=(u+I_{m^{\prime}B^{n}})\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)(x)$

for every $0\leq t\leq 1$ and $m^{\prime}\geq m$. ∎

A final approximation procedure, going through Corollary 4.3, proves Theorem
1.2, lifting the assumption (31) in Theorem 4.2. We rewrite the statement for
the convenience of the reader.

######  Theorem.

Consider $u\in\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R}^{n})$ such that
$o\in\mathrm{int}(\mathrm{dom}(u))$, and choose $q>0$. Consider
$v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ such that
$o\in\mathrm{dom}(v)$. Then

$\displaystyle\left.\frac{d}{dt}\mu_{q}(\mathrm{epi}(u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)))\right|_{t=0^{+}}=$
$\displaystyle\int_{\mathbb{R}^{n}}v^{*}(\nabla u(x))e^{-u(x)}|x|^{q-n}\,$
$\displaystyle
dx+\int_{\partial\mathrm{dom}(u)}h_{\mathrm{dom}(v)}(N_{\mathrm{dom}(u)}(y))e^{-u(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y).$

###### Proof of Theorem 1.2.

Again, our strategy is to use Lemma 3.12 to prove our claim by approximation.
Consider $u$ as in the hypotheses. We approximate $u$ through the sequence
$u_{m}\coloneqq u+I_{mB^{n}}$. By [5, Proposition 4.2]
$u_{m}\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)$ epi-
converges to $u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)$
for every $t\geq 0$. Moreover,
$u_{m}\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)\leq
u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)$ for every
$m$. Therefore, for $\varepsilon>0$ suitably small we can find $A,c>0$ such
that

$e^{u_{m}\square(t\mathbin{\vbox{\hbox{\rule{0.90417pt}{0.90417pt}}}}v)}\leq
e^{-u\square(t\mathbin{\vbox{\hbox{\rule{0.90417pt}{0.90417pt}}}}v)(x)}\leq
Ae^{-c|x|}$ (36)

for every $x\in\mathbb{R}^{n},m\in\mathbb{N}$, and $t\in[0,\varepsilon]$.
Without loss of generality, we suppose $o\in\mathrm{epi}(v)$ and
$\max_{y\in\mathrm{dom}(v)}v(y)=0$ as in Remark 3.4, so that

$h_{K^{u_{m}}}+t\overline{v^{*}}=h_{K^{u_{m}}}+th_{K^{v}}.$

This choice affects the calculation with a multiplicative constant that we can
ignore by Remark 3.10. For the sake of conciseness, denote
$u_{t}=u\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)$ and
$u_{m,t}=u_{m}\square(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v)$.

By (36)

$\displaystyle\int_{\mathbb{R}^{n}}e^{-u(x)}|x|^{q-n}\,dx-\int_{\mathbb{R}^{n}}e^{-u_{m}(x)}|x|^{q-n}\,dx=$
$\displaystyle\int_{\mathbb{R}^{n}\setminus
mB^{n}}e^{-u(x)}|x|^{q-n}\,dx\leq\int_{\mathbb{R}^{n}\setminus
mB^{n}}Ae^{-c|x|}|x|^{q-n}\,dx\to 0$

as $m\to\infty$. Thus $\mu(\mathrm{epi}(u_{m}))\to\mu(\mathrm{epi}(u))$.

We now want to prove that the derivative in $t$ of
$\mu(\mathrm{epi}(u_{m,t}))$ converges uniformly to

$\int_{\mathbb{R}^{n}}v^{*}(\nabla
u(x))e^{-u(x)}|x|^{q-n}\,dx+\int_{\partial\mathrm{dom}(u)}h_{\mathrm{dom}(v)}(N_{\mathrm{dom}(u)}(y))e^{-u(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)$

for $t\in[0,\varepsilon]$, where $\varepsilon>0$ is small enough for Corollary
4.3 to hold as well. We do so approximating the two integrals separately by
the ones in (32) evaluated for $u_{m}$ with the perturbation
$v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ (that is, $v^{*}\in
C_{rec}(\mathbb{R}^{n})$).

We start with the integral on $\mathbb{R}^{n}$. Notice that since
$o\in\mathrm{epi}(v)$, we have $v^{*}\geq 0$. By Lemma 4.6, and since
$\mathrm{dom}(u_{m,t})\subset mB^{n}+\mathrm{dom}(v)$ for $m$ suitably large,

$\displaystyle\left|\int_{\mathbb{R}^{n}}v^{*}(\nabla
u_{t}(x))e^{-u_{t}(x)}|x|^{q-n}\,dx-\int_{\mathbb{R}^{n}}v^{*}(\nabla
u_{m,t}(x))e^{-u_{m,t}(x)}|x|^{q-n}\,dx\right|$
$\displaystyle\leq\left|\int_{\mathbb{R}^{n}}v^{*}(\nabla
u_{t}(x))e^{-u_{t}(x)}|x|^{q-n}\,dx-\int_{\delta(m)B^{n}}v^{*}(\nabla
u_{m,t}(x))e^{-u_{m,t}(x)}|x|^{q-n}\,dx\right|$
$\displaystyle+\left|\int_{(mB^{n}+\mathrm{dom}(v))\setminus\delta(m)B^{n}}v^{*}(\nabla
u_{m,t}(x))e^{-u_{m,t}(x)}|x|^{q-n}\,dx\right|$
$\displaystyle\leq\int_{\mathbb{R}^{n}\setminus\delta(m)B^{n}}v^{*}(\nabla
u_{t}(x))e^{-u_{t}(x)}|x|^{q-n}\,dx+\int_{\mathbb{R}^{n}\setminus\delta(m)B^{n}}v^{*}(\nabla
u_{m,t}(x))e^{-u_{m,t}(x)}|x|^{q-n}\,dx$

By Lemma 4.5, this last line converges to $0$ independently of $t$, since by
(36) we can use (34) or (35) depending on $q$ for both the integrals there.

We now repeat a similar procedure for the integral on the boundary of
$\partial\mathrm{dom}(u)$. If
$u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, we have nothing to prove
since for $m$ sufficiently large, we obtain exactly the desired quantity. If
$\mathrm{dom}(u)=\mathbb{R}^{n}$ (and thus
$\mathrm{dom}(u_{t})=\mathbb{R}^{n}$) then

$\int_{\partial\mathrm{dom}(u_{m,t})}h_{\mathrm{dom}(v)}(N_{\mathrm{dom}(u_{m,t})}(y))e^{-u(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)=0$

converges uniformly to $0$ by the estimates following (33). If
$\mathrm{dom}(u)\neq\mathbb{R}^{n}$, then for $m$ sufficiently large
$\partial\mathrm{dom}(u_{t})\cap\partial\mathrm{dom}(u_{m,t})\neq\emptyset$.
Notice, moreover, that
$\partial\mathrm{dom}(u_{m,t})\setminus\partial\mathrm{dom}(u_{t})\subset\mathbb{R}^{n}\setminus
mB^{n}$ and
$\partial\mathrm{dom}(u_{t})\setminus\partial\mathrm{dom}(u_{m,t})\subset\mathbb{R}^{n}\setminus
mB^{n}$ for every $m$. We infer, following the same strategy in (33) and
denoting by $L$ the Lipschitz constant of $v^{*}$,

$\displaystyle\left|\int_{\partial\mathrm{dom}(u_{t})}h_{\mathrm{dom}(v)}(N_{\mathrm{dom}(u_{t})}(y))e^{-u_{t}(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)\right.-$
$\displaystyle\left.\int_{\partial\mathrm{dom}(u_{m,t})}h_{\mathrm{dom}(v)}(N_{\mathrm{dom}(u_{m,t})}(y))e^{-u_{m,t}(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)\right|$
$\displaystyle\leq\int_{\partial\mathrm{dom}(u_{t})\setminus\partial\mathrm{dom}(u_{m,t})}h_{\mathrm{dom}(v)}(N_{\mathrm{dom}(u_{t})}(y))e^{-u_{t}(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)$
$\displaystyle+\int_{\partial\mathrm{dom}(u_{m,t})\setminus\partial\mathrm{dom}(u_{t})}h_{\mathrm{dom}(v)}(N_{\mathrm{dom}(u_{m,t})}(y))e^{-u_{m,t}(y)}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)$
$\displaystyle\leq
2LAc\int_{m}^{\infty}e^{-c\tau}\int_{\partial\mathrm{dom}(u)\cap\tau
B^{n}}|y|^{q-n}\,d\mathcal{H}^{n-1}(y)d\tau.$

If $0<q<n$ we continue by

$\ldots\leq 2LAc\int_{m}^{\infty}e^{-c\tau}\tau^{n-1}d\tau.$

Otherwise, if $n\leq q$, we conclude by

$\ldots\leq 2LAc\int_{m}^{\infty}e^{-c\tau}\tau^{q-1}d\tau.$

In both cases, we have uniform convergence to $0$, concluding the proof. ∎

######  Remark 4.7.

We conclude this section noting that the same method in the proof of Theorem
1.2 can be specialized to the case $q\geq n$ without requiring any assumption
on the particular position of the epigraphs of $u$ and $v$. In particular, it
is possible to recover Theorem 1.1 when
$v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$.

##  5 Functional Wulff shapes and the Fenchel-Legendre transform

In this last section, we study the connection between the Fenchel-Legendre
transform and the construction in Definition 3.5. Many that we present can be
considered folklore and can be found (in more or less explicit form) scattered
around Rockafellar’s monograph [48]. By (17), for every
$v\in\mathrm{Conv}(\mathbb{R}^{n})$

$\begin{gathered}\mathrm{epi}(v)=\bigcap_{\boldsymbol{\nu}\in\mathbb{S}^{n}_{-}}H_{\boldsymbol{\nu}}^{-}\left(h_{\mathrm{epi}(v)}(\boldsymbol{\nu})\right)=\bigcap_{\boldsymbol{\nu}\in\mathbb{S}^{n}_{-}}H_{\boldsymbol{\nu}}^{-}\left(\frac{h_{\mathrm{epi}(v)}((g(\boldsymbol{\nu}),-1))}{\sqrt{1+|g(\boldsymbol{\nu})|^{2}}}\right)\\\
=\bigcap_{x\in\mathbb{R}^{n}}H_{g^{-1}(x)}^{-}\left(\frac{h_{\mathrm{epi}(v)}((x,-1))}{\sqrt{1+|x|^{2}}}\right)=\bigcap_{x\in\mathbb{R}^{n}}H_{g^{-1}(x)}^{-}\left(\frac{v^{*}(x)}{\sqrt{1+|x|^{2}}}\right).\end{gathered}$
(37)

In general, if $f:\mathbb{R}^{n}\to\mathbb{R}\cup\\{+\infty\\}$ is proper
(thus $f^{*}$ is proper by [47, Theorem 11.1]), by the definition of the
Fenchel-Legendre transform the epigraph of $f^{*}$ is the intersection of the
epigraphs of the affine functions $x\mapsto x\cdot y-f(x),y\in\mathbb{R}^{n}$
(the supremum of a family of functions corresponds to the intersection of
their epigraphs). These epigraphs are delimited by the affine hyperplanes with
unit normal vectors $(x,-1)/\sqrt{1+|x|^{2}}$ and distance
$f(x)/\sqrt{1+|x|^{2}}$ from the origin (if $f(x)=+\infty$, the corresponding
halfspace is trivially equivalent to $\mathbb{R}^{n+1}$). Then, equivalently

$\mathrm{epi}(f^{*})=\bigcap_{x\in\mathbb{R}^{n}}H_{g^{-1}(x)}^{-}\left(\frac{f(x)}{\sqrt{1+|x|^{2}}}\right).$
(38)

This relation can be tied with the definition of Wulff-perturbation as
follows. We recall for
$u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n}),\zeta\in
C_{rec}(\mathbb{R}^{n})$, and $t\in\mathbb{R}$ the notation

$h_{u,t}=h_{K^{u}}+t\zeta.$

######  Lemma 5.1.

Let $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, $\zeta\in
C_{rec}(\mathbb{R}^{n})$, and $t\in\mathbb{R}$. Then

$\bigcap_{\boldsymbol{\nu}\in\mathbb{S}^{n}_{-}}H^{-}_{\boldsymbol{\nu}}(h_{u,t}(\boldsymbol{\nu}))=\mathrm{epi}((u^{*}+t\zeta)^{*}).$

###### Proof.

Notice that using (18),

$\displaystyle\bigcap_{\boldsymbol{\nu}\in\mathbb{S}^{n}_{-}}H^{-}_{\boldsymbol{\nu}}(h_{u,t}(\boldsymbol{\nu}))$
$\displaystyle=\bigcap_{\boldsymbol{\nu}\in\mathbb{S}^{n}_{-}}H^{-}_{\boldsymbol{\nu}}(h_{K^{u}}(\boldsymbol{\nu})+t\bar{\zeta}(\boldsymbol{\nu}))$
$\displaystyle=\bigcap_{x\in\mathbb{R}^{n}}H^{-}_{\boldsymbol{\nu}}\left(\frac{h_{K^{u}}(x,-1)}{\sqrt{1+|x|^{2}}}+t\frac{\zeta(g^{-1}(x))}{\sqrt{1+|x|^{2}}}\right)$
$\displaystyle=\bigcap_{x\in\mathbb{R}^{n}}H_{g^{-1}(x)}^{-}\left(\frac{(u^{*}+t\zeta)(x)}{\sqrt{1+|x|^{2}}}\right).$

Thanks to (38), the last line equals to $\mathrm{epi}((u^{*}+t\zeta)^{*})$,
concluding the proof. ∎

Unfortunately, it is not always the case that for a generic $\zeta\in
C_{rec}(\mathbb{R}^{n}),$ one has $\lfloor K^{u,t}\rfloor=(u^{*}+t\zeta)^{*}$.
Indeed the envelope

$\bigcap_{\boldsymbol{\nu}\in\mathbb{S}^{n}_{-}}H^{-}_{\boldsymbol{\nu}}(h_{K^{u,t}}(\boldsymbol{\nu}))$

might be such that its projection onto $e_{n+1}^{\perp}$ is not the same as
$\mathrm{dom}((u^{*}+t\zeta)^{*})$, as we show in the following example.

######  Example 5.2.

Consider the function $u\in\mathrm{Conv}_{\mathrm{coe}}(\mathbb{R})$ given by

$\displaystyle u(x)=\begin{cases}-x-\sqrt{2}\qquad&-\sqrt{2}\leq
x\leq-\sqrt{2}/2,\\\ -\sqrt{1-|x|^{2}}\qquad&-\sqrt{2}/2\leq x\leq 1,\\\
+\infty\qquad&\textit{otherwise.}\end{cases}$

$x$$y$

$x$$y$

Figure 1: On the left, the body $K^{u}$. On the right, the radial plot of its
support function $h_{K^{u}}$. For a function $f$ on the sphere, its radial
plot is the map $\boldsymbol{\nu}\mapsto\boldsymbol{\nu}f(\boldsymbol{\nu})$.

Then $K^{u}\in\mathcal{K}^{2}$ (on the left in Figure 1) is such that its
support function (on the right in Figure 1) is, in polar coordinates for
$\theta\in[-\pi,\pi]$,

$\displaystyle h_{K^{u}}(\theta)=\begin{cases}\qquad
1\qquad&-\frac{3}{4}\pi\leq\theta\leq\frac{3}{4}\pi,\\\
{\frac{\sqrt{2}}{\sqrt{1+(\tan\theta)^{2}}}}\qquad&\text{otherwise}.\end{cases}$

Notice that $K^{u}$ can be determined by considering only the half-spaces
corresponding to $\theta\in[-3\pi/4,3\pi/4]$. Indeed, if we define

$\tilde{K}\coloneq\bigcap_{\theta\in[-3\pi/4,3\pi/4]}H^{-}_{(\cos\theta,\sin\theta)}(h_{K^{u}}(\theta)),$

one has that

$\tilde{K}\subseteq H^{-}_{(\cos\theta,\sin\theta)}(h_{K^{u}}(\theta))$

for every $\theta\notin[-3\pi/4,3\pi/4]$. Therefore, $\tilde{K}=K^{u}$.

Consider $\overline{\zeta}\in C(\mathbb{S}^{1})$ such that it is symmetric
with respect to the horizontal axis, $\overline{\zeta}(\theta)\equiv 0$ for
$\theta\in[-3\pi/4,3\pi/4]$, and is strictly positive otherwise. Then, by
construction, for every $t\geq 0$

$K^{u}\subset
H^{-}_{(\cos\theta,\sin\theta)}(h_{K^{u}}(\theta)+t\overline{\zeta}(\theta)),$

and therefore

$K^{u,t}=[h_{K^{u}}+t\overline{\zeta}]=K^{u}$ (39)

for every $t\geq 0$.

Fix $x_{0}<-\sqrt{2}$, and for $t\in[0,1]$ consider the intersection of the
line $x=tx_{0}$ with the line $y=-x-\sqrt{2}$, which is the tangent to $K^{u}$
with outer unit normal vector $(-\sqrt{2}/2,-\sqrt{2}/2)$. We denote this
intersection by the point $(x_{t},y_{t})$. We now choose $\overline{\zeta}$
such that the lines determining the half-spaces
$H^{-}_{(\cos\theta,\sin\theta)}(h_{K^{u}}(\theta)+t\overline{\zeta}(\theta)),\theta\notin[-3\pi/4,3\pi/4],$
intersect $x=tx_{0}$ in $(x_{t},y_{t})$, while the remaining ones are
unchanged. By explicit calculations, this corresponds to the choice (see
Figure 2 on the left)

$\displaystyle\overline{\zeta}(\theta)=\begin{cases}\qquad
0\qquad\qquad&-\frac{3}{4}\pi\leq\theta\leq\frac{3}{4}\pi,\\\
\frac{|(\sqrt{2}+x_{0})(|\cot\theta|-1)|}{\sqrt{1+(\cot\theta)^{2}}}\qquad&\textit{otherwise.}\end{cases}$

$x$$y$

$x$$y$

Figure 2: On the left, the radial plot of $h_{K^{u}}+t\overline{\zeta}$ (in
red, the original profile of the perturbed part). On the right, the epigraph
of $(u^{*}+t\zeta)$.

If we consider the envelope

$\bigcap_{\theta\in(-\pi,0)}H^{-}_{\boldsymbol{\nu}}(h_{K^{u}}(\theta)+t\bar{\zeta}(\theta)),$

by construction we obtain an epigraph whose corresponding function agrees with
$u$ in $[-\sqrt{2},1]$, with an additional part corresponding to the epigraph
of $y=-x-\sqrt{2}+I_{[x_{0},-\sqrt{2}]}(x)$ (see Figure 2 on the right).
Considering

$\zeta(x)=\overline{\zeta}(g^{-1}(x))\sqrt{1+(g^{-1}(x))^{2}},$

by Lemma 5.1 and (39) we just proved that, in this example,

$\lfloor K^{u,t}\rfloor\neq(u^{*}+t\zeta)^{*}.$

In order to identify well-behaved couples $(u,\zeta)$, we introduce the
following property.

######  Definition 5.3.

Fix $\zeta\in C_{rec}(\mathbb{R}^{n})$ and consider $t$ in an interval
$I\subset\mathbb{R}$. We say that the couple $(u,\zeta)$ has the property
$(\mathbf{P})$ in $I$ if

$(u^{*}+t\zeta)^{*}=\lfloor K^{u,t}\rfloor$ ($\mathbf{P}$)

for every $t\in I$.

As we mentioned before, this is always satisfied if
$\zeta=v^{*},v\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ and $t\geq 0$.
Small non-convex perturbations of the previous case provide further examples.
Another sufficient condition is the following.

######  Lemma 5.4.

Consider $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, and suppose that
$\mathrm{dom}(u)$ has non-empty interior. If $\zeta\in C_{c}(\mathbb{R}^{n})$
then, for $t$ sufficiently close to $0$ the couple $(u,\zeta)$ satisfies
$(\mathbf{P})$.

###### Proof.

We use the notation $u_{t}=(u^{*}+t\zeta)^{*}$. We start noting the following
general fact: For every $x\in\mathrm{dom}(\lfloor K^{u,t}\rfloor)$

$\lfloor K^{u,t}\rfloor(x)=u_{t}(x).$

Indeed, by construction, $K^{u,t}$ is obtained as the intersection of
$\mathrm{epi}(u_{t})$ and a suitable translation parallel to $e_{n+1}$ of
reflection of the same set with respect to $H$. The same construction shows
that, in general,

$\lfloor K^{u,t}\rfloor(x)\geq u_{t}(x)$

for every $x\in\mathbb{R}^{n}$. Notice that this implies

$\mathrm{dom}(\lfloor K^{u,t}\rfloor)\subset\mathrm{dom}(u_{t}).$

Our next claim is that, if $\zeta\in C_{c}(\mathbb{R}^{n})$, then

$\mathrm{dom}(\lfloor K^{u,t}\rfloor)=\mathrm{dom}(u_{t})=\mathrm{dom}(u)$
(40)

for $t$ sufficiently close to $0$, thus proving the lemma. Consider indeed
$x\in\partial\mathrm{dom}(u)$. Then, $X_{0}\coloneqq(x,u(x))\in\partial
K^{u}$, and there exists $\overline{\boldsymbol{\nu}}\in N_{K^{u}}(X_{0})$
such that

$g(\overline{\boldsymbol{\nu}})\cap\mathrm{supp}\ \zeta=\emptyset.$ (41)

Moreover, $X_{0}\in H^{-}_{\boldsymbol{\nu}}(h_{K^{u}}(\boldsymbol{\nu}))$ for
every $\boldsymbol{\nu}\in\mathbb{S}^{n}$. In particular, for every open
neighborhood $A$ of $\overline{N}$ there exists $\varepsilon>0$ such that
$X_{0}\in H^{-}_{\boldsymbol{\nu}}(h_{K^{u}}(\boldsymbol{\nu})+\tau)$ for
every $|\tau|<\varepsilon$ and $\boldsymbol{\nu}\in(\mathbb{S}^{n}\setminus
A)$, since $X_{0}\in\partial K^{u}$. By (41), $\mathrm{supp}\
(\overline{\zeta})$ is included in the complement of an open neighborhood of
$\overline{\boldsymbol{\nu}}$. Thus, there exists $\varepsilon>0$ such that
for every $t$ sufficiently small $|t\zeta|<\varepsilon$, and $X_{0}\in
K^{\mathrm{epi}(u_{t})}$ for every such $t$. Therefore, $u(x)\leq\lfloor
K^{u,t}\rfloor(x)$ for every $x\in\mathrm{dom}(u)$. In particular,
$\mathrm{dom}(u)\subset\mathrm{dom}\lfloor K^{u,t}\rfloor$. To prove the
claim, notice that since $\mathrm{supp}\ (\zeta)$ is compact

$\mathrm{epi}(u_{t})\subset\bigcap_{\mathbb{R}^{n}\setminus\mathrm{supp}\
(\zeta)}H^{-}_{g^{-1}(y)}(h_{u,t}(g^{-1}(y)))\subset I_{\mathrm{dom}(u)}+c$

for some $c\in\mathbb{R}$. Therefore,
$\mathrm{dom}(u_{t})\subset\mathrm{dom}(u)$. The claim now follows observing
that

$\mathrm{dom}(u_{t})\subset\mathrm{dom}(u)\subset\mathrm{dom}\lfloor
K^{u,t}\rfloor\subset\mathrm{dom}(u_{t}),$

concluding the proof. The uniform choice of an interval for $t$ comes from the
fact that $\mathbb{S}^{n}\cap H$ and $\mathrm{supp}\ {\overline{\zeta}}$ are
strictly separated. ∎

The following is a further scenario in which $(\mathbf{P})$ is satisfied. In
[10], a similar hypothesis was introduced to define admissible perturbations.

######  Lemma 5.5.

Consider $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$ and $\zeta\in
C_{rec}(\mathbb{R}^{n})$. If for every $t$ in an interval $I$ we have that
$u^{*}+t\zeta$ is convex, then the couple $(u,\zeta)$ satisfies $(\mathbf{P})$
for $t\in I$.

###### Proof.

If $u^{*}+t\zeta$ is convex, then since $\zeta\in C_{rec}(\mathbb{R}^{n})$ we
have $(u^{*}+t\zeta)^{*}\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$, and
therefore there exist bodies $K_{t}\in\mathcal{K}^{n}$ such that
$h_{K_{t}}(x,-1)=u^{*}(x)+t\zeta(x)$ for every $t\in I$. Then, by (17)
$(u^{*}+t\zeta)^{*}=\lfloor K_{t}\rfloor$. Since

$K_{t}=[h_{K_{t}}]=[h_{K^{u}}+t\bar{\zeta}]=K^{u,t},$

we infer $(u^{*}+t\zeta)^{*}=\lfloor K^{u,t}\rfloor$, which is precisely the
required property. ∎

Let us provide a practical example (communicated to us by Mussnig [46]) where
$\zeta$ is non-convex, but $u^{*}+t\zeta$ is.

######  Example 5.6.

Fix $c>0$ and consider the function $u(x)=c|x|+I_{B^{n}}(x)$. Its Fenchel-
Legendre transform is given by

$\displaystyle u^{*}(x)=\begin{cases}0\qquad\quad&|x|\leq c,\\\
|x|-c\quad&|x|\geq c.\end{cases}$

Now, take $\zeta\in C_{rec}(\mathbb{R}^{n})$ defined as

$\displaystyle\zeta(x)=\begin{cases}|x|-c\quad&|x|\leq c,\\\
0\qquad\quad&|x|\geq c,\end{cases}$

and consider the functional Wulff shape of $u$ at time $t$ with respect to
$\zeta$, taking $t\in[0,1]$. Notice that if we take the function
$v(x)=(|x|-c)+I_{B_{c}(0)}(x)$, then

$u^{*}+t\zeta=u^{*}\wedge tv,$

and by the properties of the Fenchel-Legendre transform

$u_{t}=(u^{*}+t\zeta)^{*}=(u^{*}\wedge
tv)^{*}=u\vee(t\mathbin{\vbox{\hbox{\rule{1.29167pt}{1.29167pt}}}}v^{*}).$

Therefore,

$\displaystyle u_{t}(x)=\begin{cases}ct\qquad\quad&|x|\leq t,\\\
c|x|\qquad\,&t\leq|x|\leq 1,\\\ +\infty\qquad&|x|>1.\end{cases}$

In practice, the perturbation cuts the cone, which is the graph of $u$,
orthogonally with respect to its axis, creating a plateau. As $t$ increases,
so do the height and width of the plateau.

We conclude by noting that, as proved in [29, Proposition 1.5], the Fenchel-
Legendre transform satisfies a semigroup property analog to the one in Theorem
3.8. In particular, for $u\in\mathrm{Conv}_{\mathrm{cd}}(\mathbb{R}^{n})$,
$\zeta\in C_{rec}(\mathbb{R}^{n})$, and $s,t\geq 0$

$((u^{*}+t\zeta)^{*}+s\zeta)^{*}=(u^{*}+(s+t)\zeta)^{*}.$

That is, property $(\mathbf{P})$ is not needed in order to achieve the
semigroup property.

### Acknowledgments

The core of this work originated from the Ph.D. thesis of the author,
supervised by Gabriele Bianchi and Paolo Gronchi, whose support and guidance
were fundamental. We extend our gratitude to Monika Ludwig, Fabian Mussnig,
Andrea Colesanti, and Donmeng Xi for the many fruitful discussions.
The author was supported, in part, by Austrian Science Fund (FWF):
10.55776/P34446.

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Jacopo Ulivelli

Institut für Diskrete Mathematik und Geometrie

TU Wien

Wiedner Hauptstraße 8-10/1046

1040 Wien, Austria

e-mail<EMAIL_ADDRESS>