# Persistent Intrinsic Volumes

David Cohen-Steiner  Centre INRIA d’Université Côte d’Azur, Valbonne, France
Antoine Commaret  Centre INRIA d’Université Côte d’Azur, Valbonne, France
Laboratoire Jean-Alexandre Dieudonné, Université Côte d’Azur, Nice, France

###### Abstract

We develop a new method to estimate the area, and more generally the intrinsic
volumes, of a compact subset $X$ of $\mathbb{R}^{d}$ from a set $Y$ that is
close in the Hausdorff distance. This estimator enjoys a linear rate of
convergence as a function of the Hausdorff distance under mild regularity
conditions on $X$. Our approach combines tools from both geometric measure
theory and persistent homology, extending the noise filtering properties of
persistent homology from the realm of topology to geometry. Along the way, we
obtain a stability result for intrinsic volumes.

Keywords — Geometric Measure Theory, Intrinsic Volumes, Lipschitz-Killing
Curvatures, Quermassintegrals, Persistence Homology, Geometric Inference.

MSC Classification — 53C65, 28A75, 55N31, 49Q15.

##  1 Introduction

Geometric inference deals with the retrieval of geometric characteristics of
an unknown shape $X$ from a finite sample of $X$. Among such geometric
characteristics, the _intrinsic volumes_ $V_{0}(X),\dots,V_{d}(X)$ of a subset
$X$ of a Euclidean space $\mathbb{R}^{d}$ are of particular importance.
Indeed, these quantities enjoy general properties such as additivity, isometry
invariance and positive homogeneity, making them relevant in various fields
such as integral geometry, where intrinsic volumes are known as
_quermassintegrals_ , or in differential geometry where they are called
_Lipschitz-Killing curvatures_. Notably, the codimension 1 intrinsic volume
$V_{d-1}(X)$ is a multiple of the boundary area of $X$, whose estimation is an
important question in geometric inference. It  is of practical relevance in a
large swath of scientific and industrial domains. For example, biologists seek
to measure the area of tissues such as lungs [27], bones [28], blood vessels
[20] or brains [38]. Applications in other fields such as chemistry, geology,
engineering are so numerous that they are difficult to summarize. From a
mathematical perspective, we want to understand under which conditions and at
which rate we are able to retrieve the intrinsic volumes of $X$ from an
approximating set $Y$. In order to achieve this, we introduce a novel method
to estimate the intrinsic volumes of $X$ using techniques from persistent
homology and geometric measure theory. This allows to weaken the regularity
conditions required in the current literature. Furthermore, we obtain rates of
convergence with respect to the Hausdorff distance $d_{H}(X,Y)$, allowing $X$
to be approximated by point clouds or voxels.

Intrinsic volumes were first defined for convex subsets of $\mathbb{R}^{d}$
[39, 6] and sets whose boundary is a smooth submanifold of $\mathbb{R}^{d}$
[40]. In either case, while $V_{d}(X)=\mathcal{H}^{d}(X)$ coincides with the
volume of $X$, the remaining intrinsic volumes $V_{i}(X)_{0\leq i\leq d-1}$
depend on the curvature of its boundary. If $X$ is an open set of
$\mathbb{R}^{d}$ bounded by a hypersurface, denote by $(\kappa_{i}(x))_{1\leq
i\leq d-1}$ its principal curvatures at $x\in\partial X$. Then, for all $0\leq
i\leq d-1$:

$V_{i}(X)=\frac{1}{(d-i)\omega_{d-i}}\int_{\partial
X}\Sigma_{d-i-1}(\kappa_{1},\dots,\kappa_{d-1})d\omega(x)$

where $\Sigma_{i}$ is the elementary symmetric polynomial in $d-1$ variables
of degree $i$, $\omega_{i}$ the $i$-th volume of the unit ball in dimension
$i$ and $d\omega$ the canonical volume form of $\partial X$. In particular, up
to multiplicative constants, $V_{d-1}(X)$ is the area of $\partial X$,
$V_{d-2}(X)$ is the integral of the mean curvature and $V_{0}(X)$ is the
integral of the Gauss curvature which by the Gauss-Bonnet theorem is the Euler
characteristic $\chi(X)$ of $X$. Similar formulas exist when $X$ is a
hypersurface of dimension $k$, in which case $V_{k}(X)$ is a multiple of its
area. When $X$ is a convex polytope, the $i$-th intrinsic volume is the sum of
the volumes of its $i$-dimensional facets multiplied by their appropriately
normalized exterior cone angle.

The problem of extending the definition of intrinsic volumes to broader
classes of sets was first posed by Milnor [33]. Since then, consistent
definitions were given for Whitney-stratified sets [8], sets in a o-minimal
structure [4], and general unions of sets with positive reach [42] using
respectively stratified Morse theory, o-minimal axioms and geometric measure
theory. Fu [24] gave general conditions for sets to have well-defined
intrinsic volumes and showed that those conditions hold for subanalytic sets.
Most of these definitions employ the theory of normal cycles developed in
[41], which uses the theory of currents [23]. In our framework we give a
definition of intrinsic volumes for offsets of sets with positive $\mu$-reach
without using the language of currents, the $\mu$-reach being a relaxed
version of the reach of Federer [22] (see Definition 2.2).

#### Previous work

Previous results focused mostly on the estimation of the area $V_{d-1}(X)$.
One approach is to estimate the area by the area of a piecewise linear
recontruction of the data. For example, the tangent complex triangulation [5]
guarantees that this estimator converges to $V_{d-1}(X)$ at a linear rate in
the Hausdorff distance $d_{H}(X,Y)$ when $X$ is a smooth submanifold of
$\mathbb{R}^{d}$ and $Y$ a noise-free sample. Using Crofton’s formula, [1]
builds an estimator for the surface area $\mathcal{H}^{d-1}(\partial X)$ from
a point cloud sample $Y$ and obtains a square-root rate of convergence
$O(d_{H}(X,Y)^{1/2})$ in the general case. Other works have focused on the
retrieval of curvatures measures, which are local, more informative versions
of the intrinsic volumes. A convergence rate of $d_{H}(X,Y)^{1/2}$ for the
curvature measures was obtained in [11] under the condition
$d_{H}(X,Y)\leq\operatorname{\mathrm{reach}}(X)$. Chazal et al. [14] obtain
the convergence of the curvature measures of $Y^{r}$ to that of $X^{r}$ at a
square root rate, for any fixed $r>0$, where
$Z^{r}=\\{x\in\mathbb{R}^{d}\operatorname{\,|\,}d_{Z}(x)\leq r\\}$ is the
$r$-offset of any subset $Z$ of $\mathbb{R}^{d}$.

Linear convergence rates were obtained for the estimation of the first
intrinsic volume using persistent homology of height functions and Crofton’s
formula. Authors in [16] showed a linear rate of convergence of $V_{1}(Y)$ to
$V_{1}(X)$ with respect to the Fréchet distance between $X$ and $Y$ when they
are both compact surfaces of $\mathbb{R}^{3}$ or both curves in
$\mathbb{R}^{d}$ assuming their total absolute curvature is bounded. Building
on these ideas, Edelsbrunner et al. [21] obtained more recently a linear rate
of convergence for the first intrinsic volume of voxelizations of smooth sets,
in addition to showing the convergence of all intrinsic volumes of spheres
voxelized with ever increasing precision.

Another line of research has focused on non-deterministic geometric inference
from uniform samples of convex sets. Notably, the authors of [7, 36] worked
with convex sets with a $C^{k}$ boundary, where $k\geq 2$. The expected
intrinsic volumes of the convex hull of the sample were shown to converge each
to the ideal intrinsic volume at a rate of $C_{X}n^{-2/(d+1)}$ where $n$ is
the number of sample points and $C_{X}$ a constant depending on $X$. It was
also proven that this result does not hold with a mere $C^{1}$ boundary
condition, suggesting that finding an estimator that is robust to the lack of
regularity is difficult.

#### Contributions

In this paper, we define quantities $V^{\varepsilon}_{i}(Y)$ depending on a
parameter $\varepsilon$, that approach $V_{i}(X)$ at a linear rate in
$d_{H}(X,Y)$ assuming only mild regularity conditions on $X$. This rate is
easily seen to be optimal. To the best of our knowledge, these are the first
estimators that come with theoretical guarantees beyond sets with positive
reach. Even for the basic problem of estimating the boundary area of
3-dimensional object with reach zero, we are not aware of any other provably
correct method.

######  Theorem (Main Results).

Let $X,Y$ be two compact sets of $\mathbb{R}^{d}$ and let
$\mu\in(0,1],\varepsilon>0$ be such that
$d_{H}(X,Y)\leq\varepsilon\leq\frac{1}{4}\operatorname{\mathrm{reach}}_{\mu}(X)$.
Then we have:

$\left\lvert V^{\varepsilon}_{i}(Y)-V_{i}(X^{2\varepsilon})\right\rvert\leq
C_{d}K(X^{2\varepsilon})\frac{\varepsilon}{\mu},$ (1)

where $C_{d}$ is a constant depending on $d$,
$K(X^{2\varepsilon})\coloneqq\mathcal{H}^{d}(X^{2\varepsilon})+\mathcal{H}^{d-1}(\operatorname{\mathrm{Nor}}(X^{2\varepsilon}))$
and $\operatorname{\mathrm{Nor}}(X^{2\varepsilon})$ is the unit normal bundle
of $X^{2\varepsilon}$.

Further assuming that $\operatorname{\mathrm{reach}}(X)>0$, we prove a linear
rate of convergence for the intrinsic volumes of offsets:

$\left\lvert V_{i}(X)-V_{i}(X^{2\varepsilon})\right\rvert\leq
C_{d}\left(K(X)+K(X^{2\varepsilon})\right)\frac{\varepsilon}{\mu}.$ (2)

It is worth noting that the second claim holds even when
$\operatorname{\mathrm{reach}}(X)$ is arbitrarily smaller than $\varepsilon$,
a case for which, to the best of our knowledge, no quantitative convergence
result between the curvatures of $X^{2\varepsilon}$ and $X$ was known. We also
conjecture that this claim holds when $X$ is subanalytic. Taken together, the
two claims above provide a way to estimate the intrinsic volumes of an unknown
shape $X$ with positive reach from a Hausdorff approximation $Y.$ From this
point of view, the condition that $\operatorname{\mathrm{reach}}(X)$ is
positive is not restrictive since sets with positive reach form a dense family
of compact subsets for the Hausdorff distance.

A byproduct of our methods is an answer to the second open question asked by
Milnor in [33]: In which sense do $X$ and $Y$ have to be close to guarantee
that their intrinsic volumes are close? It turns out that the existence of a
$C^{0}$-controlled homotopy equivalence (see [9] for a related notion) is
sufficient, assuming a bound on the volume of the unit normal bundle of both
sets. More precisely, say that $X$ and $Y$ are _$(\varepsilon,\delta)$
-homotopy equivalent_ if there exist two continuous maps $f:X\to Y$, $g:Y\to
X$ such that ${\left\lvert\left\lvert
f-\operatorname{\mathrm{Id}}\right\rvert\right\rvert_{\infty}\leq\varepsilon}$,
$\left\lvert\left\lvert
g-\operatorname{\mathrm{Id}}\right\rvert\right\rvert_{\infty}\leq\varepsilon$
and such that there exist homotopies $H_{1}$ (resp. $H_{2}$) between $f\circ
g$ and $\operatorname{\mathrm{Id}}_{Y}$ (resp. $g\circ f$ and
$\operatorname{\mathrm{Id}}_{X}$) satisfying $\left\lvert\left\lvert
H_{1}(t,\cdot)-\operatorname{\mathrm{Id}}_{Y}\right\rvert\right\rvert_{\infty}\leq
2\delta$ (resp. $\left\lvert\left\lvert
H_{2}(t,\cdot)-\operatorname{\mathrm{Id}}_{X}\right\rvert\right\rvert_{\infty}\leq
2\delta$) for all $t\in[0,1]$.

######  Theorem 1.1.

Let $X$ and $Y$ be two compact subsets of $\mathbb{R}^{d}$ with positive
reach. If $X$ and $Y$ are $(\varepsilon,\delta)$-homotopy equivalent for
$\varepsilon$ and $\delta$ positive, we have:

$\left\lvert V_{i}(X)-V_{i}(Y)\right\rvert\leq
C_{d}\max(\varepsilon,\delta)\left(K(X)+K(Y)\right).$ (3)

For our inference problem, a naive approach would be to estimate the intrinsic
volumes of $X$ by the intrinsic volumes of small offsets of $Y.$ However, this
leads to a trade-off between the bias induced by too large offset parameters
and the spurious geometric details that come with small offset parameters.
Optimizing this trade-off yields a sublinear rate of convergence. We use the
noise-filtering properties of persistent homology to improve over this
sublinear behavior by studying the inclusion $Y^{\varepsilon}\subset
Y^{3\varepsilon}$ instead, similar to the usual method for estimating Betti
numbers [13, 17]. Our approach uses the principal kinematic formula from
integral geometry to express the intrinsic volumes as integrals of certain
Euler characteristics. We then define our persistent intrinsic volumes by
replacing these Euler characteristics with persistent Euler characteristics
associated with the pair $Y^{\varepsilon}\subset Y^{3\varepsilon}$. A
stability theorem for image persistence then allows us to prove a linear rate
of convergence for our estimators.

## Outline

  * •

In Section 2, we describe some basic properties of distance functions and we
introduce our regularity conditions.

  * •

In Section 3, we define the intrinsic volumes, and the class of sets admitting
a normal bundle. Geometric measure theory provides the principal kinematic
formula which serves as the link between the topological and geometric
properties of a compact set admitting a normal bundle.

  * •

In Section 4 and Section 5, we give some background on persistent homology and
prove a stability theorem for image persistence modules associated with
sublevel set filtrations of non necessarily smooth functions. We also prove a
bound on the number of bars in an image persistence diagram.

  * •

In Section 6, we state and prove our main results.

  * •

In Section 7, we discuss the computational tractability of persistent
intrinsic volumes and we give a Monte-Carlo based algorithm to approximate
them.

##  2 Distance functions and $\mu$-reach of compact sets

In this section, we give definitions pertaining to non-smooth analysis and
distance functions [15].

######  Definition 2.1.

Let $X\subset\mathbb{R}^{d}$ be a closed set.

  * •

The distance to $X$ is the map $d_{X}:x\in\mathbb{R}^{d}\mapsto\inf_{z\in
X}\left\lvert\left\lvert z-x\right\rvert\right\rvert$. When $X=\\{x\\}$ is a
singleton, we write $d_{X}=d_{x}$ to ease notations. We let
$\Delta(X)\coloneqq d_{X}(0)$.

  * •

The set of _closest points to $x$ in $X$_ is denoted by $\Gamma_{X}(x)$.

  * •

The _Clarke gradient_ of $d_{X}$ at any point $x\notin X$ is the convex hull
of directions opposite to the closest points to $x$, see Figure 1.

$\operatorname{\partial^{*}\\!}d_{X}(x)\coloneqq\operatorname{\mathrm{Conv}}\left(\left\\{\frac{x-z}{\left\lvert\left\lvert
x-z\right\rvert\right\rvert}\,\Bigr{|}\,z\in\Gamma_{X}(x)\right\\}\right).$

Figure 1: Clarke gradient of $d_{X}$ outside of $X$.

The quantity $\Delta\\!\left(\operatorname{\partial^{*}\\!}d_{X}(x)\right)$
coincides with the norm of the generalized gradient
$\left\lvert\left\lvert\nabla d_{X}(x)\right\rvert\right\rvert$ defined by
Lieutier in [30], which is classically used in the definition of the
$\mu$-reach of a compact set.

######  Definition 2.2 (Reach and $\mu$-reach of a subset of
$\mathbb{R}^{d}$).

Let $X\subset\mathbb{R}^{d}$ and $\mu\in(0,1]$. We define the
$\operatorname{\mathrm{reach}}$ and the $\mu$-$\operatorname{\mathrm{reach}}$
of $X$ by:

  * •

$\operatorname{\mathrm{reach}}(X)\coloneqq\sup\left\\{t>0\,|\,0<d_{X}(x)<t\implies\mathrm{card}\;\Gamma_{X}(x)=1\right\\}$,

  * •

$\operatorname{\mathrm{reach}}_{\mu}(X)\coloneqq\sup\left\\{t>0\,|\,0<d_{X}(x)<t\implies\Delta(\operatorname{\partial^{*}\\!}d_{X}(x))\geq\mu\right\\}$.

The function $\mu\mapsto\operatorname{\mathrm{reach}}_{\mu}(X)$ is non-
increasing, and
$\operatorname{\mathrm{reach}}_{1}(X)=\operatorname{\mathrm{reach}}(X)$.

We need the following theorem [19] for the specific setting where $\phi$
equals the distance function to $X$. With this choice the theorem yields that
for any compact set $X$ with $\operatorname{\mathrm{reach}}_{\mu}(X)>0$, a
sufficiently small offset of $X$ deformation retracts to $X$ itself.

######  Theorem 2.3 (Approximate inverse flow of Lipschitz functions).

Let $\phi:\mathbb{R}^{d}\to\mathbb{R}$ be a Lipschitz real-valued function and
let $a<b\in\mathbb{R}$. Assume that:

$\inf\\{\Delta(\operatorname{\partial^{*}\\!}\phi(x))\operatorname{\,|\,}x\in\phi^{-1}(a,b]\\}=\mu>0.$

Then for every $\sigma>0$, there exists a continuous function

$C^{\sigma}_{\phi}:\left\\{\begin{array}[]{ccc}[0,1]\times\phi^{-1}(\infty,b]&\to&\phi^{-1}(-\infty,b]\\\
(t,x)&\mapsto&C^{\sigma}_{\phi}(t,x)\end{array}\right.$

such that:

  * •

For any $s>t$ and $x$ such that $C^{\sigma}_{\phi}(s,x)\in\phi^{-1}(a,b]$, we
have:

$\phi(C^{\sigma}_{\phi}(s,x))-\phi(C^{\sigma}_{\phi}(t,x))\leq-(s-t)(b-a);$

  * •

For any $t\in[0,1]$, $x\in\phi^{-1}(\infty,a]$ implies
$C^{\sigma}_{\phi}(t,x)=x$;

  * •

For any $x\in\phi^{-1}(-\infty,b]$, the map $s\mapsto C^{\sigma}_{\phi}(s,x)$
is $\frac{b-a}{\mu-\sigma}$-Lipschitz.

In particular, $C^{\sigma}_{\phi}(1,\cdot)$ is a deformation retraction
between $\phi^{-1}(-\infty,a]$ and $\phi^{-1}(-\infty,b]$.

##  3 Intrinsic Volumes and normal bundles

### Background on Intrinsic Volumes

Intrinsic volumes of subsets of $\mathbb{R}^{d}$ are geometric quantities of
importance in geometric measure theory. They were first defined by Steiner
[39] for compact convex sets and by Weyl [40] for smooth submanifolds. Federer
[22] extended their definition to the class of sets with _positive reach_.
Specifically, he proved the _tube formula_ which states that the volume of
small offsets of a set with positive reach is a polynomial whose coefficients
provide a definition for the intrinsic volumes, generalizing the definitions
of Steiner and Weyl:

$\forall\;0\leq
r\leq\operatorname{\mathrm{reach}}(X),\;\mathcal{H}^{d}(X^{r})\eqqcolon\sum_{i=0}^{d}\omega_{i}V_{d-i}(X)r^{i},$

where for any $s\geq 0$, $\mathcal{H}^{s}$ denotes the $s$-Hausdorff measure
of $\mathbb{R}^{d}$. The polynomial in the right handside is called _the
Steiner polynomial of X_ and we denote it by $Q_{X}$.

######  Proposition 3.1 (Properties of intrinsic volumes).

Let $X,Y\subset\mathbb{R}^{d}$ be two sets of positive reach. The intrinsic
volumes $V_{i}$ satisfy the following properties for all $0\leq i\leq d$:

(Isometry Invariance)

For any isometry $g:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $X$ and $g(X)$ have the
same intrinsic volumes.

(Hausdorff Continuity)

The restriction of $V_{i}$ to the class of convex subsets of $\mathbb{R}^{d}$
is continuous in the Hausdorff metric;

(Additivity)

If both $X\cup Y$ and $X\cap Y$ have positive reach, we have:

$V_{i}(X\cup Y)+V_{i}(X\cap Y)=V_{i}(X)+V_{i}(Y).$

The celebrated Hadwiger theorem [26, 29] states that intrinsic volumes form a
basis of the functionals satisfying these three properties over the class of
compact convex subsets of $\mathbb{R}^{d}$. The additivity property allows in
particular to extend the definition of intrinsic volumes to e.g finite unions
of convex sets, using the inclusion-exclusion principle.

The principal kinematic formula provides an extension of the tube formula when
the offset parameter goes beyond the reach:

######  Theorem 3.2 (Particular case of the Principal Kinematic Formula).

Let $X\subset\mathbb{R}^{d}$ be a compact set with positive reach. Then for
any $r>0$,

$\int_{\mathbb{R}^{d}}\chi(X\cap B(x,r))\mathrm{d}x=Q_{X}(r).$

Intrinsic volumes have been extended to other classes of sets while preserving
the properties and results listed above, notably the class subanalytic sets
[24], the $U_{PR}$ class i.e generic unions of sets with positive reach [42]
and the WDC class [25]. We will use another extension of intrinsic volumes,
namely to the class of sets admitting a normal bundle, which we define in the
following section.

### Sets admitting a normal bundle

We introduce a class of sets defined so as to contain small tubular
neighborhoods of sets with positive $\mu$-reach for any $\mu\in(0,1]$. We
recall [23] that the _tangent cone_ $\operatorname{\mathrm{Tan}}(X,x)$ of a
set $X\subset\mathbb{R}^{d}$ at a point $x\in X$ is defined as the cone
spanned by the set of limits of sequences of the form
$(x_{n}-x)/\left\lvert\left\lvert x_{n}-x\right\rvert\right\rvert$ where
$x_{n}$ is a sequence in $X$ converging to $x$ and not equal to $x$ for all
$n\in\mathbb{N}$. When $X$ has positive reach, the _normal cone_ of $X$ at $x$
is
$\operatorname{\mathrm{Nor}}(X,x)\coloneqq\operatorname{\mathrm{Tan}}(X,x)^{\text{o}}$
where for any set $S\subset\mathbb{R}^{d}$, its dual cone is
$S^{\text{o}}\coloneqq\\{u\in\mathbb{R}^{d}\operatorname{\,|\,}\langle
u,v\rangle\leq 0,\,\forall\,v\in S\\}.$

######  Definition 3.3 (Sets admitting a normal bundle).

Let $X\subset\mathbb{R}^{d}$. When
${}^{\neg}X\coloneqq\overline{\mathbb{R}^{d}\setminus X}$ has positive reach
and is a Lipschitz domain, define

$\operatorname{\mathrm{Nor}}(X,x)\coloneqq-\operatorname{\mathrm{Nor}}(^{\neg}X,x).$

This definition is consistent in case both $X,^{\neg}X$ have positive reach.
If either $\operatorname{\mathrm{reach}}(X)>0$ or both
$\operatorname{\mathrm{reach}}(^{\neg}X)>0$ and $X$ is a Lipschitz domain, we
say that _$X$ admits a normal bundle $\operatorname{\mathrm{Nor}}(X)$_ with

$\operatorname{\mathrm{Nor}}(X)\coloneqq\bigcup_{x\in\partial
X}\\{x\\}\times(\operatorname{\mathrm{Nor}}(X,x)\cap\mathbb{S}^{d-1}).$

The pair $(x,n)\in\operatorname{\mathrm{Nor}}(X)$ is said to be _regular_ when
$\operatorname{\mathrm{Tan}}(\operatorname{\mathrm{Nor}}(X),(x,n))$ is a
$(d-1)$-dimensional vector space. The structure of the tangent spaces of
$\operatorname{\mathrm{Nor}}(X)$ at regular pairs defines pointwise principal
curvatures. They are essentially a generalization of the classical principal
curvatures from submanifolds of $\mathbb{R}^{d}$ to sets admitting a normal
bundle, allowing to consider integrals of curvatures (see Definition 3.5) for
non-smooth sets.

######  Proposition 3.4 (Tangent spaces of normal bundles [34]).

Let $X$ be a compact subset of $\mathbb{R}^{d}$ admitting a normal bundle
$\operatorname{\mathrm{Nor}}(X)$. Then pairs in
$\operatorname{\mathrm{Nor}}(X)$ are $\mathcal{H}^{d-1}$-almost all regular,
and for any regular pair $(x,n)\in\operatorname{\mathrm{Nor}}(X)$, there
exist:

  * •

A family $\kappa_{1},\dots,\kappa_{d-1}$ in $\mathbb{R}\cup\\{\infty\\}$
called _principal curvatures of $X$ at $(x,n)$_;

  * •

A family $b_{1},\dots,b_{d-1}\in\mathbb{R}^{d}$ of vectors orthogonal to $n$
called _principal directions_ at $(x,n)$, such that the family
$\left(\frac{1}{\sqrt{1+\kappa_{i}}^{2}}b_{i},\frac{\kappa_{i}}{\sqrt{1+\kappa_{i}^{2}}}b_{i}\right)_{1\leq
i\leq d-1}$ forms an orthonormal basis of
$\operatorname{\mathrm{Tan}}(\operatorname{\mathrm{Nor}}(X),(x,n))$.

Principal curvatures $\kappa_{i}=\kappa_{i}(x,n)$ are unique up to
permutations. When $\operatorname{\mathrm{reach}}(X)>0$, every principal
curvature $\kappa_{i}$ is larger than
$-\operatorname{\mathrm{reach}}(X)^{-1}$.

Moreover, assume that $\partial X$ is a $C^{1,1}$ hypersurface, i.e there
exists a locally Lipschitz Gauss map $x\mapsto n(x)$ of outward pointing
normals. Then $n$ is differentiable $\mathcal{H}^{d-1}$-almost everywhere and
at every point $x$ where it is differentiable the multiset of principal
curvatures at $(x,n(x))$ coincide with the multiset of eigenvalues of the
differential of $n$, i.e the classical principal curvatures.

We are now able to define the $R$-curvature mass of a set admitting a normal
bundle. This quantity will appear in the explicit convergence bounds of our
main results.

######  Definition 3.5 ($R$-curvature mass of a compact set admitting a normal
bundle).

Let $X\subset\mathbb{R}^{d}$ be a compact subset of $\mathbb{R}^{d}$ admitting
a normal bundle and let $R>0$. The $R$-curvature mass of $X$ is defined by:

$M_{R}(X)\coloneqq\int_{0}^{R}\int_{\operatorname{\mathrm{Nor}}(X)}\prod_{i=1}^{d-1}\frac{\left\lvert
1+t\kappa_{i}(x,n)\right\rvert}{\sqrt{1+\kappa_{i}(x,n)^{2}}}\operatorname{\mathrm{d}\\!}\mathcal{H}^{d-1}(x,n)\operatorname{\mathrm{d}\\!}t.$

######  Proposition 3.6 (R-Curvature mass of a set with smooth boundary).

When $X$ has a $C^{1,1}$ $(d-1)$-hypersurface as boundary, this quantity can
be expressed as an integral over $\partial X$ with its volume form
$d\omega_{\partial X}$ inherited from the canonical volume form of
$\mathbb{R}^{d}$:

$M_{R}(X)=\int_{0}^{R}\int_{\partial X}\prod_{i=1}^{d-1}\left\lvert
1+t\kappa_{i}\right\rvert d\omega_{\partial X}\operatorname{\mathrm{d}\\!}t.$

###### Proof.

If $\partial X$ is smooth (or at least $C^{1,1}$) we obtain the equality with
the definition by applying the change of variable with the bijective map
$(x,n(x))\mapsto x$ whose Jacobian is
$\prod_{i=1}^{d-1}\frac{1}{\sqrt{1+\kappa^{2}_{i}(x)}}$. ∎

######  Proposition 3.7 (Bounds on $M_{R}(X)$).

Let $X$ admit a normal bundle.
If $R\leq\operatorname{\mathrm{reach}}(X)$, we have:

$M_{R}(X)=\sum_{i=1}^{d}R^{i}\omega_{i}V_{d-i}(X).$ (4)

For any $R>0$, we have:

$M_{R}(X)\leq C_{R}(d)\mathcal{H}^{d-1}(\operatorname{\mathrm{Nor}}(X)),$ (5)

where

$C_{R}(d)\coloneqq\int_{0}^{R}(1+t^{2})^{\frac{d-1}{2}}\operatorname{\mathrm{d}\\!}t.$

###### Proof.

The equality comes from the fact that for any $0\leq
t<\operatorname{\mathrm{reach}}(X)$, $1+t\kappa_{i}\geq 0$ and thus the
coefficient in front of $t^{i-1}$ integrates over
$\operatorname{\mathrm{Nor}}(X)$ to $iV_{d-i}(X)\omega_{i}$ for any $1\leq
i\leq d$. The inequality comes from the elementary fact that
${\frac{\left\lvert 1+tx\right\rvert}{\sqrt{1+x^{2}}}\leq\sqrt{1+t^{2}}}$ for
any $t,x\in\mathbb{R}$. ∎

When $X$ is a Lipschitz domain and admits a normal bundle, the work of Rataj &
Zähle [35, Theorem 9.35] allows us to define the intrinsic volumes through the
principal kinematic formula:

######  Proposition 3.8.

Let $X$ be a set admitting a normal bundle. The quantity

$Q_{X}(r)=\int_{\mathbb{R}^{d}}\chi(X\cap
B(x,r))\operatorname{\mathrm{d}\\!}x$

is a polynomial of degree $d$ in $r$. We call this polynomial the Steiner
polynomial of $X$ and define the intrinsic volumes of $X$ from its
coefficients as in the positive reach case.

Offsets of possibly irregular sets often admit a normal bundle. Indeed, for
every $0<t<\operatorname{\mathrm{reach}}_{\mu}(X)$ we know that
${}^{\neg}(X^{t})$ is a Lipschitz domain with positive reach [10], showing
that $X^{t}$ admits a normal bundle. When $X$ is a subanalytic subset of
$\mathbb{R}^{d}$ all but a finite number of offsets $X^{t}$ have a positive
$\mu$-reach for some $\mu\in(0,1]$, see [24]. As a consequence, all but a
finite number of offsets of a subanalytic set admit a normal bundle.

##  4 Basics in persistent homology

This section gives a summary of the classical notions of persistence theory so
as to make the paper more self-contained. Further information about this topic
can be found in [12].

######  Definition 4.1 (Persistence modules).

Let $\mathbb{K}$111The results in this paper do not depend on the choice of
the field $\mathbb{K}$. be a field . A persistence module $M$ is a functor
$\mathbb{R}\to\operatorname{\mathrm{Vect}}_{\mathbb{K}}$, that is a collection
of vector spaces $(M_{t})_{t\in\mathbb{R}}$ and maps $\phi^{t}_{s}:M_{s}\to
M_{t}$ for any $s\leq t\in\mathbb{R}$, such that
$\phi^{u}_{t}\circ\phi^{t}_{s}=\phi^{u}_{s}$ for any ordered triple $s\leq
t\leq u$ in $\mathbb{R}$.

We say that a persistence module is _pointwise finite dimensional_ when
$\dim(M_{t})<\infty$ for all $t\in\mathbb{R}$. Such modules admit the
following structure theorem:

######  Proposition 4.2 (Interval decomposition of persistence modules).

Given an interval $I\subset\mathbb{R}$, $\operatorname{\mathbb{1}}_{I}$ is the
persistence module defined by

$(\operatorname{\mathbb{1}}_{I})_{t}\coloneqq\left\\{\begin{array}[]{cc}\mathbb{K}&\text{
when }t\in I\\\ 0&\text{ otherwise}\end{array}\right.$

and such that the linear maps between
$(\operatorname{\mathbb{1}}_{I})_{s}\to(\operatorname{\mathbb{1}}_{I})_{t}$
are the identity when $s<t\in I$. Let $M$ be a pointwise finite dimensional
persistence module. Then there exists a multiset $\mathcal{I}$ of intervals of
$\mathbb{R}$ such that:

$M=\bigoplus_{I\in\mathcal{I}}\operatorname{\mathbb{1}}_{I}.$

The _persistence diagram_ $\operatorname{\mathrm{dgm}}(M)$ associated to the
persistence module $M$ is the multiset of $\mathbb{R}^{2}$ whose coordinates
are the bounds the intervals decomposing $M$:

$\operatorname{\mathrm{dgm}}(M)\coloneqq\bigsqcup_{I\in\mathcal{I}}\\{(\inf
I,\sup I)\\}.$

There exists a natural distance over the space of persistence modules:

######  Definition 4.3 (Morphisms, $\delta$-interleavings and interleaving
distance).

Let $M,N$ be two persistence modules and let $\delta\geq 0$.

  * •

$M^{\delta}$ denotes the persistence module $(M_{t+\delta})_{t\in\mathbb{R}}$,
i.e $M$ shifted by $\delta$.

  * •

A morphism $j$ between two persistence modules $N$ and $M$ is a natural
transformation between functors, i.e a collection of linear maps
$(j_{t})_{t\in\mathbb{R}}:N_{t}\to M_{t}$ such that the left diagram in Figure
2 commutes.

  * •

The modules $N$ and $M$ are $\delta$-_interleaved_ when there exist two
morphisms $u,v$ respectively from $M$ to $N^{\delta}$ and from $N$ to
$M^{\delta}$ such that the right diagram in Figure 2 commutes.

  * •

The interleaving distance between $M$ and $N$ is defined as:

$d_{I}(M,N)\coloneqq\inf\\{\delta\in\mathbb{R}^{+}\operatorname{\,|\,}M\text{
and }N\text{ are }\delta\text{-interleaved}\\}.$

${{M_{s}}}$${{M_{t}}}$${{M_{s}}}$${{M_{s+\delta}}}$${{M_{s+2\delta}}}$${{N_{s}}}$${{N_{t}}}$${{N_{s}}}$${{N_{s+\delta}}}$${{N_{s+2\delta}}}$$\scriptstyle{j_{s}}$$\scriptstyle{j_{t}}$$\scriptstyle{v_{s}}$$\scriptstyle{u_{s}}$$\scriptstyle{v_{s+\delta}}$$\scriptstyle{u_{s+\delta}}$
Figure 2: Commutative diagrams in the definition of morphisms (left) and
$\delta$-interleaving (right).

######  Definition 4.4 (Bottleneck distance).

A $\delta$-matching between two persistence diagrams $D,D^{\prime}$ is a
bijective map $\gamma:C\to C^{\prime}$, between subsets of $D,D^{\prime}$ such
that for any $c\in C$,
$\left\lvert\left\lvert\gamma(c)-c\right\rvert\right\rvert_{\infty}\leq\varepsilon$,
and such that for any $(a,b)\in(D\setminus C)\cup(D^{\prime}\setminus
C^{\prime})$, $\left\lvert a-b\right\rvert\leq\varepsilon$.

The _bottleneck distance_ between two diagrams $D,D^{\prime}$ is defined as:

$d_{B}(D,D^{\prime})\coloneqq\inf\\{\delta\operatorname{\,|\,}\text{There
exists a }\delta\text{-matching between }D\text{ and }D^{\prime}\\}.$

######  Theorem 4.5 (Isometry theorem).

For any pair of persistence modules $M,N$, we have:

$d_{I}(M,N)=d_{B}(\operatorname{\mathrm{dgm}}(M),\operatorname{\mathrm{dgm}}(N)).$

##  5 Image Persistence

######  Definition 5.1 (Image Persistence).

Let $M,N$ be two persistence modules and ${f:M\to N}$ be a morphism between
them. The _image persistence module $\operatorname{\mathrm{im}}f$ is the
persistence module with vector space_
${(\operatorname{\mathrm{im}}f)_{t}=f(M_{t})}$ at value $t\in\mathbb{R}$ and
whose connecting maps
$(\operatorname{\mathrm{im}}f)_{a}\to(\operatorname{\mathrm{im}}f)_{b}$ are
the restrictions of $N_{a}\to N_{b}$ to $(\operatorname{\mathrm{im}}f)_{a}$
for every pair $a\leq b\in\mathbb{R}$.

We will only deal with image persistence modules arising in the following
situation. Let $\phi:\mathbb{R}^{d}\to\mathbb{R}$ and $A\subset
B\subset\mathbb{R}^{d}$. For every $a\in\mathbb{R}$, there is an inclusion

${A\cap\phi^{-1}(-\infty,a]=A_{a}}$${B_{a}=B\cap\phi^{-1}(-\infty,a],}$$\scriptstyle{\iota^{a}}$

which yields for every dimension $0\leq j\leq d$ a morphism of persistence
modules $\iota^{\bullet}_{j}$:

$\iota^{\bullet}_{j}:H_{j}(A_{\bullet})\to H_{j}(B_{\bullet}).$

We write
$\operatorname{\mathrm{dgm}}(\phi,A,B)=\bigsqcup_{j=0}^{d}\operatorname{\mathrm{dgm}}(\operatorname{\mathrm{im}}\,\iota^{\bullet}_{j})$
for the persistence diagram obtained by taking the direct sum of the homology
induced modules in every dimension. Its Euler characteristic
$\chi(\operatorname{\mathrm{dgm}}(\phi,A,B)(r))$ is the alternating sum of the
ranks of $\iota^{\bullet}_{j}$ at filtration value $r$.

We are now in position to prove a stability theorem for image persistence
modules associated with sublevel set filtrations of locally Lipschitz
functions.

######  Theorem 5.2 (Image persistence stability theorem).

Let $h,\tilde{h}:\mathbb{R}^{d}\to\mathbb{R}$ be two real-valued function such
that $\|h-\tilde{h}\|_{\infty}\leq\varepsilon$ and
$f:\mathbb{R}^{d}\to\mathbb{R}$ be a $\kappa$-Lipschitz function. Denote
$\tilde{X}^{a}=\tilde{h}^{-1}(-\infty,a]$ and $X^{a}=h^{-1}(-\infty,a]$.
Suppose that there exists $\mu>0$ such that on $X^{2\varepsilon}\setminus
X^{-2\varepsilon}$, $h$ is locally Lipschitz and
$\Delta(\operatorname{\partial^{*}\\!}h(x))\geq\mu$. Then we have:

$d_{B}(\mathrm{dgm}(f,\tilde{X}^{-\varepsilon},\tilde{X}^{\varepsilon}),\mathrm{dgm}(f_{|X}))\leq\frac{2\kappa\varepsilon}{\mu}.$
(6)

###### Proof.

This is an extension of the stability theorem for noisy domains from [18] to
Lipschitz functions. We adapt this proof to our setting using approximate
inverse flows of Lipschitz functions obtained in Theorem 2.3. For any
$\sigma>0$, take $C_{\sigma}(\cdot,\cdot)$ a continuous deformation retraction
between $X^{2\varepsilon}$ and $X$ given by Theorem 2.3. Let $x\in
X^{2\varepsilon}$. Each trajectory $C_{\sigma}(\cdot,x)$ is
$\frac{2\varepsilon}{\mu-\sigma}$-Lipschitz and needs at most time $1$ to send
$x$ to $X$. For every $a\in\mathbb{R}$,
$C_{\sigma}(1,\cdot):X^{2\varepsilon}_{a}\to X_{a+c}$ is a continuous map with
$c\coloneqq\frac{2\kappa\varepsilon}{\mu-\sigma}$. With the same reasoning we
obtain a continuous map $X_{a}\to X^{-2\varepsilon}_{a+c}$. The homotopies
induced by the flows yield the following commutative diagram, where the
vertical and horizontal maps are induced by inclusions:

${{H_{*}(X^{2\varepsilon}_{a})}}$${{H_{*}(X^{2\varepsilon}_{a+c})}}$${{H_{*}(X^{2\varepsilon}_{a+2c})}}$${{H_{*}(\tilde{X}^{\varepsilon}_{a})}}$${{H_{*}(\tilde{X}^{\varepsilon}_{a+c})}}$${{H_{*}(\tilde{X}^{\varepsilon}_{a+2c})}}$${{H_{*}(X_{a})}}$${{H_{*}(X_{a+c})}}$${{H_{*}(X_{a+2c})}}$${{H_{*}(\tilde{X}^{-\varepsilon}_{a})}}$${{H_{*}(\tilde{X}^{-\varepsilon}_{a+c})}}$${{H_{*}(\tilde{X}^{-\varepsilon}_{a+2c})}}$${{H_{*}(X^{-2\varepsilon}_{a})}}$${{H_{*}(X^{-2\varepsilon}_{a+c})}}$${{H_{*}(X^{-2\varepsilon}_{a+2c})}}$

As in [18], the two colored arrow paths provide interleavings showing that,
for every $\sigma\in(0,\mu)$:

$d_{B}(\operatorname{\mathrm{dgm}}(f,\tilde{X}^{-\varepsilon},\tilde{X}^{\varepsilon}),\operatorname{\mathrm{dgm}}(f_{|X}))\leq\frac{2\kappa\varepsilon}{\mu-\sigma}.$

∎

Now let $X,Y$ be two compact sets of $\mathbb{R}^{d}$ and
$f=d_{x}:z\mapsto\left\lvert\left\lvert z-x\right\rvert\right\rvert$ be the
distance function to any point $x$. Applying the previous theorem with
$h=d_{X}+2\varepsilon$, $\tilde{h}=d_{Y}+2\varepsilon$ yields the following
corollary:

######  Corollary 5.3 (Image stability theorem for compact sets).

Let $\mu\in(0,1]$, $\varepsilon>0$ and $X,Y$ be two compact subsets of
$\mathbb{R}^{d}$ such that
$d_{H}(X,Y)\leq\varepsilon\leq\frac{1}{4}\operatorname{\mathrm{reach}}_{\mu}(X)$.
Then for any $x\in\mathbb{R}^{d}$,

$d_{B}(\operatorname{\mathrm{dgm}}(d_{x},Y^{\varepsilon},Y^{3\varepsilon}),\operatorname{\mathrm{dgm}}(d_{x_{|X^{2\varepsilon}}}))\leq\frac{2\varepsilon}{\mu}.$
(7)

Now we use results from Bauer & Lesnick [2] to show that a persistence module
sandwiched between two persistence modules cannot be smaller in a certain
sense than the image persistence module. We formalize that by saying that a
persistence diagram $D^{\prime}$ injects into another persistence diagram $D$
when there is an injective map $\phi:D^{\prime}\to D$ such that
$\phi((a^{\prime},b^{\prime}))=(a,b)$ with $a\leq a^{\prime}$ and
$b^{\prime}\leq b$ for all $(a^{\prime},b^{\prime})$ in $D^{\prime}$.

######  Theorem 5.4 (Interleaving of bars in image persistence).

Let $A,B,C$ be persistent modules that are decomposable into intervals. Let
furthermore be $\varphi,\psi$ morphisms of persistent modules and write
$j=\psi\circ\varphi$:

${A}$${B}$${C}$$\scriptstyle{\varphi}$$\scriptstyle{j}$$\scriptstyle{\psi}$

Then $\operatorname{\mathrm{dgm}}(\mathrm{im}\,j)$ injects into
$\operatorname{\mathrm{dgm}}(B)$. In particular, if $B$ has a finite
decomposition in intervals, then so does $\operatorname{\mathrm{im}}j$.

###### Proof.

The morphisms of persistence modules $im(\varphi)\to B$, $im(\varphi)\to
im(j)$ are respectively monomorphism and an epimorphism of persistence
modules. By Lemma 4.2 in [2], we know that there exist injections of barcodes
$\operatorname{\mathrm{dgm}}(\operatorname{\mathrm{im}}\,\varphi)\hookrightarrow\mathcal{\operatorname{\mathrm{dgm}}}(B)$
and
$\operatorname{\mathrm{dgm}}(\operatorname{\mathrm{im}}\,j)\hookrightarrow\operatorname{\mathrm{dgm}}(\operatorname{\mathrm{im}}\,\varphi)$
respectively extending the intervals to the left and to the right. ∎

######  Remark 5.5.

For a persistence diagram $D$ and for two real numbers $a<b$, define
$N_{a}^{b}(D)$ to be the total number of bars of $D$ intersecting with
$[a,b]$. The theorem above implies that, with the same notations,
$N_{a}^{b}(\operatorname{\mathrm{dgm}}(\operatorname{\mathrm{im}}j))\leq
N_{a}^{b}(\operatorname{\mathrm{dgm}}(B))$. This is a generalization to
persistence modules of the fact that the rank of a linear map cannot exceed
the dimension of a vector space it factors through.

##  6 Persistent Intrinsic Volumes

In this section, we define the persistent intrinsic volumes
$V^{\varepsilon,R}_{i}(Y)$, where $\varepsilon,R$ are positive numbers, and
state our main results.

######  Definition 6.1 (Persistent Intrinsic Volumes).

Let $Y\subset\mathbb{R}^{d}$ be closed, $x\in\mathbb{R}^{d}$ and
$\varepsilon\geq 0$. We let
$D^{\varepsilon,x}_{Y}\coloneqq\operatorname{\mathrm{dgm}}(d_{x},Y^{\varepsilon},Y^{3\varepsilon})$.
When $\varepsilon=0$, we write
$D^{x}_{Y}\coloneqq\operatorname{\mathrm{dgm}}(d_{x_{|Y}})$.

  * •

The _persistent Steiner function_ $Q^{\varepsilon}_{Y}$ is defined by:

$Q^{\varepsilon}_{Y}(r)\coloneqq\int_{x\in\mathbb{R}^{d}}\chi(D^{\varepsilon,x}_{Y}(r))\mathrm{d}x.$

  * •

Given $R>0$, the persistent Steiner polynomial is the orthogonal projection of
$Q^{\varepsilon}_{Y}$ restricted to $[0,R]$ on the space of polynomials of
degree at most $d$ for the scalar product of $L^{2}\left([0,R]\right)$.

  * •

Writing the persistent Steiner polynomial as
$\sum_{i=0}^{d}\omega_{i}V^{\varepsilon,R}_{d-i}(Y)r^{i}$, the
rescaled coefficients $(V^{\varepsilon,R}_{i}(Y))_{0\leq i\leq d}$ are the
_persistent intrinsic volumes_ of $Y$.

######  Remark 6.2 (Persistent intrinsic volumes are well-defined).

When $Y$ is compact and $\varepsilon$ is positive, one can always find a
finite simplicial complex $C$ such that $Y^{\varepsilon}\subset C\subset
Y^{3\varepsilon}$. By Theorem 5.4 the diagrams $D^{\varepsilon,x}_{Y}$ inject
into $D^{x}_{C}$ for every $x$, implying that their size is bounded by the
number of simplices of $C$. In particular, the persistent Steiner function is
locally bounded, and persistent intrinsic volumes are well-defined without any
regularity condition on $Y$.

The following is an immediate consequence of Corollary 5.3 and Theorem 5.4.

######  Proposition 6.3 (Diagram approximation).

Let $X,Y$ be compact subsets of $\mathbb{R}^{d}$ and $\varepsilon,\mu>0$ be
such that
$d_{H}(X,Y)\leq\varepsilon\leq\frac{1}{4}\operatorname{\mathrm{reach}}_{\mu}(X)$.
For all $x\in\mathbb{R}^{d}$, we have:

  * •

$d_{B}(D^{\varepsilon,x}_{Y},D^{x}_{X^{2\varepsilon}})\leq\frac{2\varepsilon}{\mu}$;

  * •

$D^{\varepsilon,x}_{Y}$ injects into $D^{x}_{X^{2\varepsilon}}$.

The next lemma bounds the average difference of the Euler characteristics of
close persistent diagrams.

######  Lemma 6.4 ($\chi$-averaging lemma).

Let $D,D^{\prime}$ be two homology persistent diagrams with
$d_{B}(D,D^{\prime})\leq\varepsilon$. Then for any $a<b\in\mathbb{R}$ we have:

$\int_{a}^{b}\left\lvert\chi(D(t))-\chi(D^{\prime}(t))\right\rvert\mathrm{d}t\leq
2\varepsilon(N_{a}^{b}(D)+N_{a}^{b}(D^{\prime})).$ (8)

If $D^{\prime}$ injects into $D$, we have:

$\int_{a}^{b}\left\lvert\chi(D(t))-\chi(D^{\prime}(t))\right\rvert\mathrm{d}t\leq
2\varepsilon N_{a}^{b}(D).$ (9)

###### Proof.

The first inequality is a slight extension of an argument obtained in [16].
Let $\mathcal{I}_{i}$ (resp. $\mathcal{I}^{\prime}_{i}$) be the set of
intervals of the decomposition of $D$ (resp. $D^{\prime}$) in dimension $i$.
We have:

$\chi(D(t))=\sum_{i=0}^{d}(-1)^{i}\sum_{I_{i}\in\mathcal{I}_{i}}\operatorname{\mathbb{1}}_{I_{i}}(t).$

Let $\gamma$ be an $\varepsilon$-matching between $D$ and $D^{\prime}$. Define
$C_{i}$ and $C^{\prime}_{i}$ to be the respective largest subsets of
$\mathcal{I}_{i}$ and $\mathcal{I}^{\prime}_{i}$ matched bijectively by
$\gamma$. We have:

$\displaystyle\int_{a}^{b}\left\lvert\chi(D(t))-\chi(D^{\prime}(t))\right\rvert\operatorname{\mathrm{d}\\!}t\leq\sum_{i=0}^{d}$
$\displaystyle\Bigl{(}\sum_{I_{i}\in
C_{i}}\int_{a}^{b}\left\lvert\operatorname{\mathbb{1}}_{I_{i}}-\operatorname{\mathbb{1}}_{\gamma(I_{i})}\right\rvert\operatorname{\mathrm{d}\\!}t$
$\displaystyle+\sum_{I_{i}\in\mathcal{I}_{i}\setminus
C_{i}}\int_{a}^{b}\operatorname{\mathbb{1}}_{I_{i}}(t)\operatorname{\mathrm{d}\\!}t+\sum_{I^{\prime}_{i}\in\mathcal{I}^{\prime}_{i}\setminus
C^{\prime}_{i}}\int_{a}^{b}\operatorname{\mathbb{1}}_{I^{\prime}_{i}}(t)\operatorname{\mathrm{d}\\!}t\Bigr{)}.$

Since $\gamma$ is an $\varepsilon$-matching, each of these integrals is
bounded by $2\varepsilon$. Now for the first term in the right-hand side, the
support of the map
$\left\lvert\operatorname{\mathbb{1}}_{I_{i}}-\operatorname{\mathbb{1}}_{\gamma(I_{i})}\right\rvert$
is included in $I_{i}\cup\gamma(I_{i})$, meaning that its integral over
$[a,b]$ vanishes if none of these intervals intersect with $(a,b)$. As for the
remaining terms, only intervals intersecting with $[a,b]$ contribute to the
sum. Overall, there is as most one non-vanishing contribution per interval in
$D\cup D^{\prime}$ whose intersection with $(a,b)$ is non-empty, and no
contribution otherwise. The total number of non-vanishing integrals is bounded
by $N_{a}^{b}(D)+N_{a}^{b}(D^{\prime})$.

In case $D^{\prime}$ injects into $D$, there is a $\varepsilon$-matching such
that $C^{\prime}_{i}=\mathcal{I}^{\prime}_{i}$ and such that
$\gamma(I_{i})\subset I_{i}$ for every interval $I_{i}$ of $D$, meaning there
is at most one non-vanishing contribution per interval of $D$ whose
intersection with $(a,b)$ is non-empty, leading to the desired bound
$2\varepsilon N_{a}^{b}(D)$. ∎

We use the previous lemmas to establish a linear rate of convergence of
$Q^{\varepsilon}_{Y}$ to $Q_{X^{2\varepsilon}}$ over $[0,R]$ for any positive
real $R$.

######  Theorem 6.5 (Estimating the Steiner polynomial in the $L^{1}$ norm).

Let $X,Y\subset\mathbb{R}^{d}$ be compact sets and $\varepsilon,\mu>0$ be such
that
$d_{H}(X,Y)\leq\varepsilon\leq\frac{1}{4}\operatorname{\mathrm{reach}}_{\mu}(X)$.
Then we have:

$\left\lvert\left\lvert
Q^{\varepsilon}_{Y}-Q_{X^{2\varepsilon}}\right\rvert\right\rvert_{L^{1}([0,R])}\leq\frac{4\varepsilon}{\mu}\int_{\mathbb{R}^{d}}N_{0}^{R}(D^{x}_{X^{2\varepsilon}})\mathrm{d}x.$
(10)

###### Proof.

The result is an immediate consequence of Proposition 6.3 along with the
second case in Lemma 6.4. ∎

The next lemma relates the bound of Theorem 6.7 to the $R$-curvature mass of
$X^{2\varepsilon}$.

######  Lemma 6.6 (Number of critical points of $d_{x}$).

Let $X\subset\mathbb{R}^{d}$ be a compact set and $\varepsilon,\mu>0$ be such
that $0<2\varepsilon<\operatorname{\mathrm{reach}}_{\mu}(X)$. For any $R>0$,
define
$\pi_{X^{2\varepsilon},R}:\operatorname{\mathrm{Nor}}(X^{2\varepsilon})\times[0,R]\to\mathbb{R}^{d},(y,n,t)\mapsto
y+tn$.
Then $\mathcal{H}^{d}$-almost everywhere in $x$,

  * •

$\mathrm{card}\;\pi_{X^{2\varepsilon},R}^{-1}(x)$ is finite;

  * •

$N_{0}^{R}(D^{x}_{X^{2\varepsilon}})\leq\operatorname{\mathbb{1}}_{X^{2\varepsilon}}(x)+\mathrm{card}\;\pi_{X^{2\varepsilon},R}^{-1}(x)$.

Moreover, we have:

$\int_{\mathbb{R}^{d}}N_{0}^{R}(D^{x}_{X^{2\varepsilon}})\operatorname{\mathrm{d}\\!}x\leq\operatorname{\mathrm{Vol}}(X^{2\varepsilon})+M_{R}(X^{2\varepsilon}).$

###### Proof.

The map $\pi_{X^{2\varepsilon},R}$ is Lipschitz map between two rectifiable
sets of dimension $d$. By the coarea formula of Federer [23, Theorem 3.2.22],
denoting by $J_{d}\pi_{X^{2\varepsilon},R}$ the $d$-dimensional Jacobian of
$\pi_{X^{2\varepsilon},R}$, we have:

$\int_{\mathbb{R}^{d}}\mathrm{card}\;\pi_{X^{2\varepsilon},R}^{-1}(x)\operatorname{\mathrm{d}\\!}x=\int_{\operatorname{\mathrm{Nor}}(X^{2\varepsilon})\times[0,R]}J_{d}\pi_{X^{2\varepsilon},R}(y,n,t)\operatorname{\mathrm{d}\\!}\mathcal{H}^{d}(y,n,t).$
(11)

Since $\pi_{X^{2\varepsilon},R}$ is Lipschitz on the compact set
$\operatorname{\mathrm{Nor}}(X^{2\varepsilon})\times[0,R]$, this quantity has
to be finite, yielding the finiteness of
$\mathrm{card}\;\pi_{X^{2\varepsilon},R}^{-1}(x)$ almost everywhere.

Let $x\in\mathbb{R}^{d}$ and let $X^{2\varepsilon}_{t}\coloneqq
X^{2\varepsilon}\cap d_{x}^{-1}(-\infty,t]$ be the $\sqrt{t}$-sublevel sets of
${f_{x}\coloneqq d^{2}_{x_{|X^{2\varepsilon}}}}$. When $t<0$, the set
$X^{2\varepsilon}_{t}$ is empty. When $t=0$, this set is either a point when
$x\in X^{2\varepsilon}$, or empty. The function
$f_{x}:X^{2\varepsilon}\to\mathbb{R}$ cannot be Morse in the classical sense
of Milnor [32] as $X^{2\varepsilon}$ is not a submanifold of $\mathbb{R}^{d}$.
Nevertheless by the $\mu$-reach assumption, $X^{2\varepsilon}$ is an offset of
$X$ at a regular value of its distance function, a setting to which the
generalized Morse theory of [19] applies. Specifically, if $f_{x}$ is a Morse
function in the sense of [19], the changes in homology of the filtration
$(X^{2\varepsilon}_{t})_{t>0}$ happen at the square roots of the critical
values of $f_{x}$ and there is exactly one homology change per critical point
of this map. The number of bars of a persistent homology diagram of a Morse
function is bounded by its number of critical points and we have:

$N_{0}^{R}(D^{x}_{X^{2\varepsilon}})\leq\operatorname{\mathbb{1}}_{X^{2\varepsilon}}(x)+\mathrm{card}\Bigl{\\{}\text{Critical
points of }f_{x}\text{ with distance to }x\text{ in }(0,R]\Bigr{\\}}.$ (12)

We prove that the set of points $x$ in $\mathbb{R}^{d}$ such that $f_{x}$ is a
Morse function in the sense of this generalized Morse theory has full Lebesgue
measure, and we relate the number of critical points of $f_{x}$ to the map
$\pi_{X^{2\varepsilon},R}$. Following [19, Definition 2.5], a point $z\neq x$
is a critical point for $f_{x}$ with value in $(0,R^{2}]$ if and only if the
pair $\left(z,\frac{z-x}{\left\lvert\left\lvert
z-x\right\rvert\right\rvert}\right)\eqqcolon(z,n)$ belongs to the normal
bundle $\operatorname{\mathrm{Nor}}(X^{2\varepsilon})$ and
$\left\lvert\left\lvert x-z\right\rvert\right\rvert\leq R$, that is, when
$(z,n,\left\lvert\left\lvert
x-z\right\rvert\right\rvert)\in\pi_{X^{2\varepsilon},R}^{-1}(x)$. In case
$f_{x}$ is Morse, the previous inequality becomes:

$N_{0}^{R}(D^{x}_{X^{2\varepsilon}})\leq\operatorname{\mathbb{1}}_{X^{2\varepsilon}}(x)+\mathrm{card}\;\pi_{X^{2\varepsilon},R}^{-1}(x).$
(13)

We shall prove that $\mathcal{H}^{d}$-almost everywhere in $x$, the critical
points of $f_{x}$ are non-degenerate. Let $M$ be the set of points $x$ such
that $\pi_{X^{2\varepsilon},R}^{-1}(x)$ is infinite and let
$\mathrm{Bad}(X^{2\varepsilon})$ be the set of non-regular pairs of
$\operatorname{\mathrm{Nor}}(X^{2\varepsilon})$. The set
${\mathrm{reg}(X^{2\varepsilon})\coloneqq\mathbb{R}^{d}\setminus(M\cup\pi_{X^{2\varepsilon},R}(\mathrm{Bad}_{X^{2\varepsilon}})\cup\partial
X^{2\varepsilon})}$ has full Lebesgue measure.

Take $x\in\mathrm{reg}(X^{2\varepsilon})$ and let $(b_{i})_{1\leq i\leq m}$ be
an orthonormal basis of principal directions of
$\operatorname{\mathrm{Tan}}\left(\operatorname{\mathrm{Nor}}(X^{2\varepsilon}),\left(z,\frac{z-x}{\left\lvert\left\lvert
z-x\right\rvert\right\rvert}\right)\right)$ associated with the principal
curvatures $(\kappa_{i})_{1\leq i\leq m}$. Let
$u=\sum_{i=1}^{m}\lambda_{i}b_{i}$, $v=\sum_{i=1}^{m}\nu_{i}b_{i}$ be any pair
in
$\operatorname{\mathrm{Tan}}\left(\operatorname{\mathrm{Nor}}(X^{2\varepsilon}),\left(z,\frac{z-x}{\left\lvert\left\lvert
z-x\right\rvert\right\rvert}\right)\right)$ and $r=\left\lvert\left\lvert
z-x\right\rvert\right\rvert$. Following notations from [19, Definition 2.5],
the Hessian of $f_{x}$ at $z$ is:

$(H_{|X^{2\varepsilon}}f_{x})(z):(u,v)\mapsto\sum_{i=1}^{m}(2+2r\kappa_{i})\lambda_{i}\nu_{i}.$

This bilinear form is degenerate if and only if $\displaystyle
r=-\frac{1}{\kappa}$ where $\kappa$ is a principal curvature of a point in
$\pi_{X^{2\varepsilon},R}^{-1}(x)$. There are at most $d$ distinct principal
curvatures, from which we know that the set

$\displaystyle\mathrm{Degen}(X^{2\varepsilon})\coloneqq\left\\{\Bigl{(}z,n,-\frac{1}{\kappa}\right)$
$\displaystyle\bigg{|}\;(z,n)\in\operatorname{\mathrm{Nor}}(X^{2\varepsilon})\text{
is regular,}$ $\displaystyle\kappa\text{ is a principal curvature at
}(z,n)\Bigr{\\}}.$

has Hausdorff dimension at most $(d-1)$. Thus
$\mathrm{reg}(X^{2\varepsilon})\setminus\pi_{X^{2\varepsilon},R}(\mathrm{Degen}(X^{2\varepsilon}))$
is a set of full Lebesgue measure whose elements $x$ satisfy
$N_{0}^{R}(D^{x}_{X^{2\varepsilon}})\leq\operatorname{\mathbb{1}}_{X^{2\varepsilon}}(x)+\mathrm{card}\;\pi_{X^{2\varepsilon},R}^{-1}(x)$.

Now by the structure of tangent spaces (Proposition 3.4) we have:

$J_{d}\phi(x,n,t)=\prod_{i=1}^{d-1}\frac{\left\lvert
1+t\kappa_{i}(x,n)\right\rvert}{\sqrt{1+\kappa_{i}(x,n)^{2}}}.$

and thus by the co-area formula Equation 11,

$\int_{\mathbb{R}^{d}}N_{0}^{R}(D^{x}_{X^{2\varepsilon}})\leq\int_{\mathbb{R}^{d}}\left(\operatorname{\mathbb{1}}_{X^{2\varepsilon}}(x)+\mathrm{card}\;\pi_{X^{2\varepsilon},R}^{-1}(x)\right)\operatorname{\mathrm{d}\\!}x=\operatorname{\mathrm{Vol}}(X^{2\varepsilon})+M_{R}(X^{2\varepsilon}).$
(14)

∎

We are now in position to prove our main theorems.

######  Theorem 6.7 (Linear convergence of the persistent intrinsic volumes).

Let $X,Y$ be compact subsets of $\mathbb{R}^{d}$ and let $\varepsilon,\mu>0$
be such that
$d_{H}(X,Y)\leq\varepsilon\leq\frac{1}{4}\operatorname{\mathrm{reach}}_{\mu}(X)$.
There exist constants $P(i,d)$ such that for any $R>0$, we have:

$\left\lvert
V^{\varepsilon,R}_{i}(Y)-V_{i}(X^{2\varepsilon})\right\rvert\leq\frac{\varepsilon
P(i,d)}{\mu
R^{i+1}}\left(\operatorname{\mathrm{Vol}}(X^{2\varepsilon})+M_{R}(X^{2\varepsilon})\right).$
(15)

###### Proof.

Applying Theorem 6.5 and Lemma 6.6 we obtain for any positive $R$:

$\left\lvert\left\lvert
Q^{\varepsilon}_{Y}-Q_{X^{2\varepsilon}}\right\rvert\right\rvert_{L^{1}([0,R])}\leq\frac{4\varepsilon}{\mu}(\operatorname{\mathrm{Vol}}(X^{2\varepsilon})+M_{R}(X^{2\varepsilon})).$

Let $i\leq j$ and let $(L_{n})_{n\in\mathbb{N}}$ be the Legendre polynomials
on $[0,1]$. We obtain after renormalization and reparametrization an
orthonormal basis of the polynomials of degree at most $d$ on $[0,R]$ formed
by the following polynomials:

$P^{R}_{j}(X)\coloneqq\sqrt{\frac{2j+1}{R}}L_{j}\left(\frac{X}{R}\right).$

The fact that $\left\lvert\left\lvert
L_{i}\right\rvert\right\rvert_{\infty,[0,1]}\leq 1$ yields
$\left\lvert\left\lvert
P^{R}_{j}\right\rvert\right\rvert_{\infty,[0,R]}\leq\sqrt{\frac{2j+1}{R}}$ for
any positive $R$.

Denoting by $c_{i}(P^{R}_{j})$ the $i$-th coefficient of $P^{R}_{j}$, we have
for any $i\leq j$:

$\left\lvert
c_{i}(P^{R}_{j})\right\rvert=\frac{1}{R^{i+\frac{1}{2}}}\sqrt{2j+1}\binom{j}{i}\binom{i+j}{i}$

and $c_{i}(P^{R}_{j})=0$ otherwise. Now decomposing $Q^{\varepsilon}_{Y}$ in
the basis $P^{R}_{j}$ and taking the coefficient of $X^{i}$ yields

$V^{\varepsilon,R}_{i}(Y)=\frac{1}{\omega_{d-i}}\int_{0}^{R}\int_{\mathbb{R}^{d}}\chi(D^{\varepsilon,x}_{Y}(r))\sum_{j=i}^{d}P^{R}_{j}(r)c_{i}(P^{R}_{j})\operatorname{\mathrm{d}\\!}x\operatorname{\mathrm{d}\\!}r.$

Since we have

$\left\lvert\left\lvert
P^{R}_{j}c_{i}(P^{R}_{j})\right\rvert\right\rvert_{\infty,[0,R]}\leq\frac{(2j+1)}{R^{i+1}}\binom{j}{i}\binom{i+j}{i},$

we see that:

$\left\lvert
V^{\varepsilon,R}_{i}(Y)-V_{i}(X^{2\varepsilon})\right\rvert\leq\frac{1}{R^{i+1}}\left\lvert\left\lvert
Q^{\varepsilon}_{Y}-Q_{X^{2\varepsilon}}\right\rvert\right\rvert_{L^{1}([0,R])}\frac{1}{\omega_{d-i}}\sum_{j=i}^{d}(2j+1)\binom{j}{i}\binom{i+j}{i}.$

We now obtain the desired inequality by applying Lemma 6.6 and Theorem 6.5 and
putting

$P(i,d)\coloneqq\frac{4}{\omega_{d-i}}\sum_{j=i}^{d}(2j+1)\binom{j}{i}\binom{i+j}{i}.$

∎

######  Remark 6.8.

Our approach consists in bounding the $L^{1}$ norm of
$Q^{\varepsilon}_{Y}-Q_{X^{2\varepsilon}}$ (Theorem 6.5 and Lemma 6.6):

$\left\lvert\left\lvert
Q^{\varepsilon}_{Y}-Q_{X^{2\varepsilon}}\right\rvert\right\rvert_{L^{1}([0,R])}\leq\frac{4\varepsilon}{\mu}\left(\operatorname{\mathrm{Vol}}(X^{2\varepsilon})+M_{R}(X^{2\varepsilon})\right)$

and retrieving quantities that are close to the rescaled coefficients
$(V_{i}(X^{2\varepsilon}))_{0\leq i\leq d}$ of the polynomial
$Q_{X^{2\varepsilon}}$ from the persistent Steiner function
$Q^{\varepsilon}_{Y}$, which is in $L^{2}([0,R])$ but is not a priori a
polynomial. We chose in Definition 6.1 the rescaled coefficients
$V^{\varepsilon,R}_{i}(Y)$ of its orthogonal projection on the space
$\mathbb{R}_{d}[X]$ of polynomials of degree at most $d$. From this definition
and from the explicit formulas for Legendre polynomials (see the proof of
Theorem 6.7) we obtain:

$\left\lvert
V^{\varepsilon,R}_{i}(Y)-V_{i}(X^{2\varepsilon})\right\rvert\leq\frac{P(i,d)}{4R^{i+1}}\left\lvert\left\lvert
Q^{\varepsilon}_{Y}-Q_{X^{2\varepsilon}}\right\rvert\right\rvert_{L^{1}([0,R])}.$
(16)

Taking the $i$-th coefficients of orthogonal projections is one way to build
linear forms $\phi^{R}_{i,d}:L^{2}([0,R])\to\mathbb{R}$ such that, restricted
to the space $\mathbb{R}_{d}[X]$ of polynomials with degree at most $d$,
$\phi^{R}_{i,d}$ is the map $\sum_{0\leq j\leq d}a_{j}X^{j}\mapsto a_{i}$. In
fine we defined $V_{i}^{\varepsilon,R}(Y)$ as
$\phi^{R}_{i,d}(Q^{\varepsilon}_{Y})/\omega_{d-i}$. By Equation 16 the linear
forms $\phi^{R}_{i,d}$ are $\omega_{d-i}P(i,d)/4R^{i+1}$-Lipschitz for the
$L^{1}$ norm over $[0,R]$. The bound we infer on
$V^{\varepsilon,R}_{i}(Y)-V_{i}(X^{2\varepsilon})$ from
$\left\lvert\left\lvert
Q^{\varepsilon}_{Y}-Q_{X^{2\varepsilon}}\right\rvert\right\rvert_{L^{1}([0,R])}$
comes from a bound on the Lipschitz constant of $\phi^{R}_{i,d}$ obtained
using Legendre polynomials. Yet, by the Hahn-Banach extension theorem, the
best Lipschitz constant possible for a linear form $\phi^{R}_{i,d}$ whose
restriction to $\mathbb{R}_{d}[X]$ is the $i$-th coefficient map is exactly
the Lipschitz constant of $(\phi^{R}_{i,d})_{|\mathbb{R}_{d}[X]}$ which we
denote by $l^{R}_{i,d}$. When $i=d\geq 0$ by classical work in optimization
(e.g [37, 4.9, p. 117]), we have:

$l^{R}_{d,d}=\frac{4^{d}}{R^{d+1}}\leq\frac{(2d+1)}{R^{d+1}}\binom{2d}{d}$

where the right handside is the bound on the Lipschitz constant we explicitly
obtained by the Legendre polynomials method. This shows that there exist
alternate ways of defining the persistent intrinsic volumes from the
persistent Steiner function that lead to a strictly better bound than the one
in Theorem 6.7.

######  Theorem 6.9 (Convergence of the intrinsic volumes of an offset).

Let ${X\subset\mathbb{R}^{d}}$ and $\mu,\varepsilon>0$ be such that
$\varepsilon<\frac{1}{2}\operatorname{\mathrm{reach}}_{\mu}(X)$. If
$\operatorname{\mathrm{reach}}(X)>0$, we have:

$\left\lvert V_{i}(X)-V_{i}(X^{2\varepsilon})\right\rvert\leq\frac{\varepsilon
P(i,d)}{\mu
R^{i+1}}\left(\operatorname{\mathrm{Vol}}(X)+\operatorname{\mathrm{Vol}}(X^{2\varepsilon})+M_{R}(X)+M_{R}(X^{2\varepsilon})\right).$

When $X$ is subanalytic, this inequality holds when we replace $M_{R}(X)$ by
$\displaystyle\liminf_{r\to 0}M_{R}(X^{r})$.

###### Proof.

Let $x\in\mathbb{R}^{d}$ and $\delta>0$. For any sufficiently small
$\sigma>0$, set $c=\frac{2\varepsilon-\delta}{\mu-\sigma}$. By the $\mu$-reach
hypothesis, there exists a continuous flow between $X^{2\varepsilon}$ and
$X^{\delta}$ which is $c$-Lipschitz in the time parameter thanks to Theorem
2.3. This yields the following commutative diagram:

${{H_{*}(X^{2\varepsilon}_{a})}}$${{H_{*}(X^{2\varepsilon}_{a+c})}}$${{H_{*}(X^{2\varepsilon}_{a+2c})}}$${{H_{*}(X^{\delta}_{a})}}$${{H_{*}(X^{\delta}_{a+c})}}$${{H_{*}(X^{\delta}_{a+c})}}$

This gives a $\frac{2\varepsilon-\delta}{\mu-\sigma}$ interleaving between the
two persistence modules, which implies
$d_{B}(D^{x}_{X^{2\varepsilon}},D^{x}_{X^{\delta}})\leq\frac{2\varepsilon}{\mu}$
by letting $\sigma$ go to zero. Using the same reasoning as in the proof of
Theorem 6.7, except that we have to use the first inequality of Lemma 6.4
since a priori there is no injection between the two diagrams $D^{x}_{X}$ and
$D^{x}_{X^{2\varepsilon}}$, we get for any positive $R$:

$\left\lvert
V_{i}(X^{2\varepsilon})-V_{i}(X^{\delta})\right\rvert\leq\frac{\varepsilon
P(i,d)}{\mu
R^{i+1}}\int_{\mathbb{R}^{d}}(N_{0}^{R}(D^{x}_{X^{2\varepsilon}})+N_{0}^{R}(D^{x}_{X^{\delta}}))\operatorname{\mathrm{d}\\!}x.$
(17)

By Lemma 6.6:

$\left\lvert
V_{i}(X^{2\varepsilon})-V_{i}(X^{\delta})\right\rvert\leq\frac{\varepsilon
P(i,d)}{\mu
R^{i+1}}\left(\operatorname{\mathrm{Vol}}(X^{2\varepsilon})+M_{R}(X^{2\varepsilon})+\operatorname{\mathrm{Vol}}(X^{\delta})+M_{R}(X^{\delta})\right).$
(18)

We already know that $\operatorname{\mathrm{Vol}}(X^{\delta})$ converges to
$\operatorname{\mathrm{Vol}}(X)$. Let us prove the first statement and assume
that $\operatorname{\mathrm{reach}}(X)>0$. By the tube formula, the intrinsic
volumes of $X^{\delta}$ are the (scaled) coefficients of the Steiner
polynomial of $X$ translated by $\delta$, and thus $V_{i}(X^{\delta})$
converges to $V_{i}(X)$ when $\delta$ goes to zero. We are left to prove that
$\lim_{\delta\to 0}M_{R}(X^{\delta})=M_{R}(X)$ to obtain the desired
inequality. If $0\leq\delta<\operatorname{\mathrm{reach}}(X)$, writing
$h_{\delta}:(x,n)\mapsto(x+\delta n,n)$, we have:

$\operatorname{\mathrm{Nor}}(X^{\delta})=\bigcup_{(x,n)\in\operatorname{\mathrm{Nor}}(X)}\\{x+\delta
n\\}\times\\{n\\}=h_{\delta}(\operatorname{\mathrm{Nor}}(X)).$

Denoting by $\kappa_{i}(z,n)$ the principal curvatures of $X$ at a regular
pair $(z,n)$ of $X$ and $\kappa_{\delta,i}(z^{\prime},n^{\prime})$ the
principal curvatures of $X^{\delta}$ at a regular pair
$(z^{\prime},n^{\prime})$ of $\operatorname{\mathrm{Nor}}(X^{\delta})$, we
have:

$\kappa_{\delta,i}(z+\delta n,n)=f_{\delta}(\kappa_{i}(z,n)),$

where $f_{\delta}(s)=\frac{s}{1+\delta s}$. Since
$\kappa_{i}(x,n)\geq-\frac{1}{\operatorname{\mathrm{reach}}(X)}$ the
quantities $f_{\delta}(\kappa_{i}(x,n))$ are well-defined for any
$\delta<\operatorname{\mathrm{reach}}(X)$. The change of variable formula
yields for any positive $t$:

$\displaystyle M_{R}(A^{\delta})=$
$\displaystyle\int_{\operatorname{\mathrm{Nor}}(A^{\delta})}\prod_{i=1}^{d-1}\frac{\left\lvert
1+t\kappa_{\delta,i}\right\rvert}{\sqrt{1+\kappa^{2}_{i,\delta}}}\operatorname{\mathrm{d}\\!}\mathcal{H}^{d-1}(x,n)$
$\displaystyle=$
$\displaystyle\int_{\operatorname{\mathrm{Nor}}(A)}J_{d-1}(h_{\delta})\prod_{i=1}^{d-1}\frac{\left\lvert
1+tf_{\delta}(\kappa_{i})\right\rvert}{\sqrt{1+f_{\delta}(\kappa_{i})^{2}}}\operatorname{\mathrm{d}\\!}\mathcal{H}^{d-1}(x,n).$

Since $h_{\delta}-\operatorname{\mathrm{Id}}$ is $\delta$-Lipschitz, we have

$(1-\delta)^{d-1}\leq J_{d-1}(h_{\delta}(x,n))\leq(1+\delta)^{d-1}.$

Hence, as $\delta$ tends to 0, $J_{d-1}h_{\delta}$ tends to $1$ and
$f_{\delta}(\kappa)$ tend to $\kappa$ for all $\kappa\in\mathbb{R}$. Lebesgue
dominated convergence theorem gives:

$\displaystyle\lim_{\delta\to 0}M_{R}(X^{\delta})=$
$\displaystyle\int_{0}^{R}\int_{\operatorname{\mathrm{Nor}}(A)}\prod_{i=1}^{d-1}\frac{\left\lvert
1+t\kappa_{i}\right\rvert}{\sqrt{1+\kappa_{i}^{2}}}\operatorname{\mathrm{d}\\!}\mathcal{H}^{d-1}(x,n)\operatorname{\mathrm{d}\\!}t$
$\displaystyle=$ $\displaystyle\;M_{R}(X).$

For the second claim, assuming $X$ is subanalytic, we make use of the theory
of normal cycles of subanalytic sets developed in [24]. The function $d_{X}$
is a subanalytic aura of $X$ and as such we have convergence of the normal
cycles $N_{X^{\delta}}$ of $X^{\delta}$ to that of $X$ for the flat norm as
$\delta$ tends to 0. In particular, as intrinsic volumes are the normal cycles
integrated against Lipschitz-Killing forms, we have for every $0\leq i\leq
d-1$, $\displaystyle\lim_{\delta\to 0}V_{i}(X^{\delta})=V_{i}(X)$ and the last
claim is a consequence of Equation 18 as $\delta$ tends to 0. ∎

Combining Theorem 6.9 and Theorem 6.7 gives us a way to estimate $V_{i}(X)$
from $Y$:

######  Corollary 6.10 (Linear rate of approximation for the intrinsic
volumes).

Let $X,Y\subset\mathbb{R}^{d}$ and $\mu,\varepsilon>0$ be such that
$d_{H}(X,Y)\leq\varepsilon<\frac{1}{4}\operatorname{\mathrm{reach}}_{\mu}(X)$.
If ${\operatorname{\mathrm{reach}}(X)>0}$, we have:

$\left\lvert
V_{i}(X)-V^{\varepsilon,R}_{i}(Y)\right\rvert\leq\frac{\varepsilon P(i,d)}{\mu
R^{i+1}}\left(\operatorname{\mathrm{Vol}}(X)+2\operatorname{\mathrm{Vol}}(X^{2\varepsilon})+M_{R}(X)+2M_{R}(X^{2\varepsilon})\right).$
(19)

We prove Theorem 1.1 using similar methods as before, except that the
interleavings between persistence modules stem from the existence of a
$(\varepsilon,\delta)$-homotopy equivalence.

###### Proof of Theorem 1.1.

Let $x$ be any point of $\mathbb{R}^{d}$ and let $M_{a}\coloneqq M\cap B(x,a)$
for any subset $M$ of $\mathbb{R}^{d}$ and $a\in\mathbb{R}$. The map
$H_{1}:[0,1]\times Y\to Y$ is a homotopy between $f\circ g$ and
$\operatorname{\mathrm{Id}}_{Y}$, and by assumption its restriction to $Y_{a}$
is a continuous map $H^{a}_{1}:[0,1]\times Y_{a}\to Y_{a+2\delta}$. This
yields a homotopy between $f\circ g:Y_{a}\to Y_{a+2\delta}$ and the inclusion
$Y_{a}\xhookrightarrow{}Y_{a+2\delta}$. Letting $\delta^{\prime}$ be a
positive real such that $2\varepsilon+2\delta^{\prime}\geq 2\delta$, we obtain
this commutative diagram of continuous maps:

${{X_{a+\varepsilon}}}$${{X_{a+\varepsilon+\delta^{\prime}}}}$${{Y_{a}}}$${{Y_{a+2\varepsilon+\delta^{\prime}}}}$${{Y_{a+2\varepsilon+2\delta^{\prime}}}}$$\scriptstyle{g}$$\scriptstyle{\psi}$$\scriptstyle{f}$$\scriptstyle{\phi}$

Since the same holds symmetrically for $H^{a}_{2}$, we can apply the homology
functor to obtain the following commutative diagram thanks to the homotopy
between $\psi\circ\phi$ (resp. $\phi\circ\psi$) and the inclusion
$Y_{a}\xhookrightarrow{}Y_{a+2\varepsilon+2\delta^{\prime}}$ (resp.
$X_{a}\xhookrightarrow{}X_{a+2\varepsilon+2\delta^{\prime}}$):

${{H_{*}(X_{a})}}$${{H_{*}\left(X_{a+\varepsilon+\delta^{\prime}}\right)}}$${{H_{*}\left(X_{a+2\varepsilon+2\delta^{\prime}}\right)}}$${{H_{*}(Y_{a})}}$${{H_{*}\left(Y_{a+\varepsilon+\delta^{\prime}}\right)}}$${{H_{*}\left(Y_{a+2\varepsilon+2\delta^{\prime}}\right)}}$

Optimizing on $\delta^{\prime}$, this yields
$d_{B}(D^{x}_{X},D^{x}_{Y})\leq\max(\varepsilon,\delta)$ for any
$x\in\mathbb{R}^{d}$. Now following the same line of reasoning as the proof of
Theorem 6.9, first bounding the $L^{1}$ norm of $Q_{X}-Q_{Y}$, and then
retrieving its coefficients, we obtain the bound:

$\left\lvert
V_{i}(X)-V_{i}(Y)\right\rvert\leq\frac{\max(\varepsilon,\delta)P(i,d)}{R^{i+1}}\left(\operatorname{\mathrm{Vol}}(X)+M_{R}(X)+\operatorname{\mathrm{Vol}}(Y)+M_{R}(Y)\right).$

∎

######  Remark 6.11.

The bounds obtained in Theorem 6.7, Theorem 6.9, Corollary 6.10 and Theorem
1.1 depend on the parameter $R$ and explode when $R\to 0$ or $\infty$. While
this is not critical for the asymptotic rate, it could be interesting to find
ways of guessing good values of $R$ a priori. The main results stated in the
introduction are obtained taking $R=1$ and using Proposition 3.7.

##  7 Computing persistent intrinsic volumes

We conclude this paper by discussing how one may compute our estimators in
practice, assuming $Y$ is given as a finite set of points. From the proof of
Theorem 6.7 the $V^{\varepsilon,R}_{i}(Y)$ are given by:

$V^{\varepsilon,R}_{i}(Y)=\int_{0}^{R}\int_{\mathbb{R}^{d}}\chi(D^{\varepsilon,x}_{Y}(r))S_{i,d}(r)\operatorname{\mathrm{d}\\!}x\operatorname{\mathrm{d}\\!}r,$

where $S_{i,d}$ is a polynomial.

Computing this integral exactly would involve computing a $d$-dimensional
family of persistence diagrams, which is computationally daunting. We may
however easily approximate $V^{\varepsilon,R}_{i}(Y)$ with arbitrary accuracy
using a Monte-Carlo method by sampling points $x$ uniformly in the $R$-offset
of $Y$ and computing the image persistence diagrams of their distance function
$d_{x}$ using e.g [3]. The number of trials required to reach accuracy
$\delta$ is of the order of $\delta^{-2}$ times the variance of the random
variable $\displaystyle
V=\int_{0}^{R}\chi(D^{\varepsilon,x}_{Y}(r))S_{i,d}(r)\operatorname{\mathrm{d}\\!}r$.

The variance of $V$ is bounded by $\sup\left\lvert V\right\rvert^{2}$ and we
have:

$\sup\left\lvert
V\right\rvert\leq\sup_{x\in\mathbb{R}^{d}}N_{0}^{R}(D^{\varepsilon,x}_{Y})\sup_{r\in[0,R]}RS_{i,d}(r)$

that is, a polynomial in $R$ times the maximum size of the image persistence
diagram of $d_{x}$. Now by Proposition 6.3, the sizes of these image
persistence diagrams are at most the sizes of the persistence diagrams of
$d_{x_{|X^{2\varepsilon}}}$.

One situation where we can uniformly bound the size of these diagrams is when
$X$ is a semi-algebraic set, since by Thom-Milnor [31] the number of critical
points of $d_{x_{|X^{2\varepsilon}}}$ is bounded by a function of the degrees
of the polynomial equalities and inequalities defining $X$. Another bound is
provided by Remark 6.2. This bound is pessimistic in general as the smallest
simplicial complex sandwiched between $Y^{\varepsilon}$ and $Y^{3\varepsilon}$
may have a large number of simplices, yet one expects the number of critical
points of $d_{x_{|C}}$ to be much smaller on average. We leave more in-depth
investigations of the computational aspects of our method for future work.

## Acknowledgement

The authors are thankful to Mathijs Wintraecken for his advices on redacting
the present document.

## References

  * [1] Catherine Aaron, Alejandro Cholaquidis, and Ricardo Fraiman.  Estimation of surface area.  Electronic Journal of Statistics, (2), 2022.  doi:10.1214/22-EJS2031.
  * [2] Ulrich Bauer and Michael Lesnick.  Induced matchings and the algebraic stability of persistence barcodes.  Journal of Computational Geometry, March 2015.  arXiv:1311.3681, doi:10.20382/jocg.v6i2a9.
  * [3] Ulrich Bauer and Maximilian Schmahl.  Efficient computation of image persistence.  In Erin W. Chambers and Joachim Gudmundsson, editors, 39th International Symposium on Computational Geometry (SoCG 2023), Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, 2023. Schloss Dagstuhl – Leibniz-Zentrum für Informatik.  doi:10.4230/LIPIcs.SoCG.2023.14.
  * [4] Andreas Bernig.  The normal cycle of a compact definable set.  Israel J. Math., 159:373–411, 2007.  doi:10.1007/s11856-007-0052-4.
  * [5] Jean-Daniel Boissonnat and Arijit Ghosh.  Manifold reconstruction using tangential Delaunay complexes.  Discrete Comput. Geom., 51(1):221–267, 2014.  doi:10.1007/s00454-013-9557-2.
  * [6] T. Bonnesen and W. Fenchel.  Theorie der konvexen Körper, volume 3 of Ergeb. Math. Grenzgeb. Springer-Verlag, Berlin, 1934.
  * [7] Károly J. Böröczky, Jr., Lars Michael Hoffmann, and Daniel Hug.  Expectation of intrinsic volumes of random polytopes.  Period. Math. Hungar., 57(2):143–164, 2008.  doi:10.1007/s10998-008-8143-4.
  * [8] Ludwig Bröcker and Martin Kuppe.  Integral geometry of tame sets.  Geom. Dedicata, 82(1-3):285–323, 2000.  doi:10.1023/A:1005248711077.
  * [9] T. A. Chapman.  Controlled simple homotopy theory and applications, volume 1009 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1983.  doi:10.1007/BFb0066066.
  * [10] Frédéric Chazal, David Cohen-Steiner, André Lieutier, and Boris Thibert.  Shape smoothing using double offsets.  In Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling - SPM ’07, 2007.  doi:10.1145/1236246.1236273.
  * [11] Frédéric Chazal, David Cohen-Steiner, and Quentin Mérigot.  Boundary measures for geometric inference.  Found. Comput. Math., 10(2):221–240, 2010.  doi:10.1007/s10208-009-9056-2.
  * [12] Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot.  The structure and stability of persistence modules.  SpringerBriefs in Mathematics. Springer, 2016.  doi:10.1007/978-3-319-42545-0.
  * [13] Frédéric Chazal and André Lieutier.  Weak feature size and persistant homology: computing homology of solids in $\mathbb{R}^{n}$ from noisy data samples.  In Computational Geometry (SCG’05), pages 255–262. ACM, New York, 2005.  doi:10.1145/1064092.1064132.
  * [14] Frédéric Chazal, David Cohen-Steiner, André Lieutier, and Boris Thibert.  Stability of curvature measures.  Computer Graphics Forum, 2009.  doi:10.1111/j.1467-8659.2009.01525.x.
  * [15] Frank H. Clarke.  Generalized gradients and applications.  Transactions of the American Mathematical Society, 205:247–262, 1975.
  * [16] David Cohen-Steiner and Herbert Edelsbrunner.  Inequalities for the curvature of curves and surfaces.  Found. Comput. Math., 7(4):391–404, 2007.  doi:10.1007/s10208-005-0200-3.
  * [17] David Cohen-Steiner, Herbert Edelsbrunner, and John Harer.  Stability of persistence diagrams.  06 2005.  doi:10.1007/s00454-006-1276-5.
  * [18] David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Dmitriy Morozov.  Persistent homology for kernels, images, and cokernels.  In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1011–1020. SIAM, Philadelphia, PA, 2009.
  * [19] Antoine Commaret.  Generalized Morse theory for tubular neighborhoods, 2024.  Preprint.  arXiv:2401.04034.
  * [20] Peter Dockery and John Fraher.  The quantification of vascular beds: A stereological approach.  Experimental and molecular pathology, 05 2007.  doi:10.1016/j.yexmp.2006.12.011.
  * [21] Herbert Edelsbrunner and Florian Pausinger.  Approximation and convergence of the intrinsic volume.  Adv. Math., 287:674–703, 2016.  doi:10.1016/j.aim.2015.10.004.
  * [22] Herbert Federer.  Curvature measures.  Transactions of the American Mathematical Society, (3), 1959.  doi:10.1090/S0002-9947-1959-0110078-1.
  * [23] Herbert Federer.  Geometric measure theory, volume Band 153 of Die Grundlehren der mathematischen Wissenschaften.  Springer-Verlag New York, Inc., New York, 1969.
  * [24] Joseph H. G. Fu.  Curvature measures of subanalytic sets.  Amer. J. Math., 116(4):819–880, 1994.  doi:10.2307/2375003.
  * [25] Joseph H. G. Fu, Dušan Pokorný, and Jan Rataj.  Kinematic formulas for sets defined by differences of convex functions.  Adv. Math., 311:796–832, 2017.  doi:10.1016/j.aim.2017.03.003.
  * [26] Hugo Hadwiger.  Vorlesungen über Inhalt, Oberfläche und Isoperimetrie.  1957\.
  * [27] Christian Karagiannidis, Stephan Strassmann, Daniel Brodie, Philine Ritter, Anders Larsson, Ralf Borchardt, and Wolfram Windisch.  Impact of membrane lung surface area and blood flow on extracorporeal CO2 removal during severe respiratory acidosis.  Intensive Care Medicine Experimental, 12 2017.  doi:10.1186/s40635-017-0147-0.
  * [28] Y-M Kim, Jae Hoon Ahn, Heung Sik Kang, and Hee Joong Kim.  Estimation of the extent of osteonecrosis of the femoral head using MRI.  Number 6. The British Editorial Society of Bone and Joint Surgery, 1998.
  * [29] D.A. Klain and G.C. Rota.  Introduction to Geometric Probability.  Introduction to Geometric Probability. Cambridge University Press, 1997.
  * [30] André Lieutier.  Any open bounded subset of $\mathbb{R}^{n}$ has the same homotopy type than its medial axis.  In Proceedings of the Eighth ACM Symposium on Solid Modeling and Applications - SM ’03, 2003.  doi:10.1145/781606.781620.
  * [31] J. Milnor.  On the Betti Numbers of Real Varieties.  Number 2. American Mathematical Society, 1964.
  * [32] John W. Milnor.  Morse theory. Based on lecture notes by M. Spivak and R. Wells, volume 51 of Ann. Math. Stud. Princeton University Press, Princeton, NJ, 1963.  doi:10.1515/9781400881802.
  * [33] J.W. Milnor.  Euler characteristic and finitely additive Steiner measures.  Collected Papers. Publish or Perish, 1994.
  * [34] Jan Rataj and Martina Zähle.  Legendrian cycles and curvatures.  J. Geom. Anal., 25(4):2133–2147, 2015.  doi:10.1007/s12220-014-9506-1.
  * [35] Jan Rataj and Martina Zähle.  Curvature measures of singular sets.  Springer Monographs in Mathematics. Springer, 2019.  doi:10.1007/978-3-030-18183-3.
  * [36] Matthias Reitzner.  Stochastical approximation of smooth convex bodies.  Mathematika, 51(1–2):11–29, 2004.  doi:10.1112/S0025579300015473.
  * [37] John R. Rice.  The Approximation of Functions. Vol. 1: Linear Theory.  1964\.
  * [38] Gretel Sanabria-Diaz, Lester Melie-Garcia, Yasser Iturria, Yasser Alemán-Gómez, Gertrudis Hernández-González, Lourdes Beatriz Valdes Urrutia, Lídice García, and Pedro Valdés-Sosa.  Surface area and cortical thickness descriptors reveal different attributes of the structural human brain networks.  NeuroImage, 05 2010.  doi:10.1016/j.neuroimage.2010.01.028.
  * [39] Jakob Steiner.  Gesammelte Werke 2.  1882\.
  * [40] Hermann Weyl.  On the Volume of Tubes.  Johns Hopkins University Press, 1939.
  * [41] M. Zähle.  Integral and current representation of Federer’s curvature measures.  Arch. Math. (Basel), 46(6):557–567, 1986.  doi:10.1007/BF01195026.
  * [42] M. Zähle.  Curvatures and currents for unions of sets with positive reach.  Geom. Dedicata, 23(2):155–171, 1987.  doi:10.1007/BF00181273.