# Local 3D Editing via 3D Distillation of CLIP Knowledge

Junha Hyung1,2† Sungwon Hwang1 Daejin Kim3 Hyunji Lee1 Jaegul Choo1

1KAIST AI 2Kakao Enterprise Corp. 3Scatter Lab
{sharpeeee, shwang.14, alee6868<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

3D content manipulation is an important computer vision task with many real-
world applications (e.g., product design, cartoon generation, and 3D Avatar
editing). Recently proposed 3D GANs can generate diverse photorealistic
3D-aware contents using Neural Radiance fields (NeRF). However, manipulation
of NeRF still remains a challenging problem since the visual quality tends to
degrade after manipulation and suboptimal control handles such as 2D semantic
maps are used for manipulations. While text-guided manipulations have shown
potential in 3D editing, such approaches often lack locality. To overcome
these problems, we propose Local Editing NeRF (LENeRF), which only requires
text inputs for fine-grained and localized manipulation. Specifically, we
present three add-on modules of LENeRF, the Latent Residual Mapper, the
Attention Field Network, and the Deformation Network, which are jointly used
for local manipulations of 3D features by estimating a 3D attention field. The
3D attention field is learned in an unsupervised way, by distilling the zero-
shot mask generation capability of CLIP to the 3D space with multi-view
guidance. We conduct diverse experiments and thorough evaluations both
quantitatively and qualitatively.111We will make our code publicly available.

Figure 1: Our local editing NeRF (LENeRF) enables users to edit specific areas
of 3D assets based on textual prompts by estimating a 3D mask for tri-plane
features. For instance, given an original 3D radiance field (a), users can
define their desired area to edit (underlined text prompt, e.g., ”eyes”).
LENeRF then generates a 3D mask, which is employed for feature fusion,
allowing for targeted modifications that adhere to the editing prompt (e.g.,
”blue eyes”) (b). Additionally, as illustrated in (c), the 3D mask itself can
be rendered and visualized for further analysis.

$\dagger$$\dagger$footnotetext: This work was done during an internship at
Kakao Enterprise Corp.

##  1 Introduction

Figure 2: Concept figure of LENeRF. Our method enables local editing of 3D
assets by generating the target feature and estimating a 3D mask which guides
the model on where to make changes at the feature level. Note that the mask is
estimated for tri-plane features, not for raw RGB outputs.

3D content editing has many real-world applications including but not limited
to product design, cartoon generation, and 3D Avatar editing. However, it
often necessitates the use of sophisticated tools with complex interfaces,
which can be difficult for novice users and labor-intensive even for seasoned
professionals. While explicit 3D representations such as voxels and meshes are
commonly used for 3D generation and editing [17, 31, 54], they are memory-
intensive and lack photorealism. In contrast, recent advances in Neural
Radiance Fields (NeRF) [33] have shown promising progress in representing 3D
environments using implicit representations [14, 21, 37, 33] combined with
volume rendering techniques that enable high-quality novel view synthesis.
NeRF-based 3D GANs [43, 36, 5, 11, 50, 52, 16, 4] have made further progress
towards generating a category of 3D aware contents with a single model,
extending the per-scene optimization scheme of NeRF.

Several studies [29, 48, 45, 44] have attempted to address the challenges of
NeRF editing, yet certain limitations persist. Works such as Edit-NeRF [29]
and CLIP-NeRF [48] have pioneered NeRF manipulations, but they are constrained
to low-resolution synthetic datasets and lack the capability to perform
localized editing. Opposed to translation [9, 53] or style transfer [13]
tasks, editing typically demands a certain degree of localization. However,
achieving this with text-only control proves to be a challenging objective.
Alternative methods [45, 44] that rely on semantic masks for editing face
their own limitations: 2D guidance is not ideal for 3D editing and lacks the
descriptiveness required for fine-grained editing. Furthermore, these
approaches require inversion steps and are difficult to generalize across
different domains, as they depend on the availability of labeled semantic
masks.

To overcome the existing limitations, we propose Local Editing NeRF (LENeRF),
which focuses on the important aspects of 3D editing: photorealism, multi-view
consistency, usability, diversity, and locality. With LENeRF, high-resolution
photo-realistic radiance fields can be edited while maintaining their quality
and multi-view consistency. One notable advantage of LENeRF is its text-only
editing, making it more usable than other methods. This allows our approach to
be applied to any domain by leveraging the multi-modal embedding space of
Contrastive Language Image Pre-training (CLIP) [39]. Additionally, our method
achieves real-time editing as it does not require any test-time optimization
process.

Our proposed approach exhibits particularly robust performance in local 3D
editing. This is achieved through a unique method of editing features in the
3D space independently by granting position-wise freedom to the features. The
naive approach of directly manipulating the latent code often results in
global changes to the 3D content, because features in the 3D space are
spatially entangled with each other as the entire radiance field is
conditioned with a single latent code. To address this issue, we propose to
generate a 3D mask on the region of interest with a masking prompt (e.g.,
”hair”) and manipulate the features inside the region while leaving the rest
unchanged. Inspired by the previous approach which introduces the explanation
method for capturing the regions of the interest [25], we estimate 3D masks in
an unsupervised fashion by using 3D distillation of the 2D CLIP model.
Although the CLIP model is not 3D-aware and the 3D GAN lacks text-conditioned
mask generation capability, our method enables the collaboration of two pre-
trained models to generate a text-conditioned 3D mask, as demonstrated in
Figure 1 (c).

LENeRF comprises three add-on modules, namely Latent Residual Mapper (LRM),
Attention Field Network (AFN), and Deformation Network (DN) as depicted in
Figure 3. LRM generates a latent code that produces a target feature field.
AFN generates a soft 3D mask indicating our region of interest. The source
feature field is distorted using DN and subsequently interpolated with the
target field to synthesize the final feature field. LENeRF is trained with
CLIP guidance [39, 38], and AFN is additionally trained with CLIP-generated
zero-shot pseudo labels.

The main contributions of our paper are as follows:

  * •

We introduce Local Editing NeRF (LENeRF), a 3D content editing framework
capable of localized, photorealistic editing using a convenient real-time
text-based interface.

  * •

Our method consists of add-on modules and does not require any domain-specific
labels, allowing the method to be generalized to other models and domains.

  * •

Our proposed technique involves a novel 3D distillation of CLIP knowledge,
specifically an unsupervised approach that utilizes the 3D GAN and CLIP models
jointly to generate 3D masks.

  * •

We present diverse quantitative and qualitative results, along with various
applications such as sequential editing, real image editing, and out-of-
distribution editing.

##  2 Related Work

3D-Aware GANs While Generative Adversarial Networks (GANs) [15] have
demonstrated their ability to generate high-quality 2D images, generating
3D-aware content with GANs remains a challenging task. Several approaches have
attempted to integrate GANs and 3D scene representations [34, 35]. Recent
progress in 3D GANs [43, 36, 5, 11, 50, 52, 16, 4, 27] that employ NeRF [33]
have achieved success in generating high-quality, multi-view consistent
images. Building upon the progress made by previous research in this field,
our work seeks to enhance its applicability by introducing a novel method for
text-guided local editing.

NeRF Manipulations To enable local editing in 3D assets, EditNeRF [29]
proposes a conditional NeRF that enables users to modify the color and shape
of a specific region using scribbles. However, the editing ability of EditNeRF
is limited to adding or removing local parts of 3D objects. Semantic mask
guided 3D editing approaches [29, 22, 46, 44] have made some progress in 3D
editing. Among these works, FENeRF[45] is trained to render 3D images and
their corresponding semantic masks, allowing editing to be performed by
inverting the model with edited semantic masks to modify 3D models. However,
using a 2D mask for 3D editing is suboptimal since fine-grained attributes
such as expression or texture cannot be manipulated using masks. Also, these
methods require segmentation maps for both training and editing, which is not
applicable to certain domains where segmentation maps are not available.
Meanwhile, there is a line of research utilizing CLIP [39] for 3D generation
and editing [32, 42, 20, 19, 3, 48, 26] that leverages the ease of
manipulation using a text. However, the generated and edited content are not
photorealistic and cannot provide fine-grained control. On the other hand, our
work focuses on high-quality, localized manipulations that are well-suited for
real-world applications.

##  3 Preliminaries

NeRF NeRF [33] is an implicit representation of 3D space using a MLP $\Psi$,
as $(\textbf{c},\sigma)=\Psi(\textbf{x},\textbf{d})$, where
$\textbf{x}=(x,y,z)$ is a point in 3D space, d is a viewing direction, and
resulting ($\textbf{c},\sigma$) are color and volume density, respectively.
The color and density values along the same ray are integrated to estimate a
single pixel given camera extrinsic and pose as:

$\hat{C}(\textbf{r})=\int_{k_{n}}^{k_{f}}T(k)\sigma(\textbf{r}(k))c(\textbf{r}(k),\textbf{d})\
dk,$ (1)

where $k_{n}$ and $k_{f}$ are near and far bounds,
$\textbf{r}(k)=\textbf{o}+k\textbf{d}$ denotes a ray projected from the camera
centered at position o, and
$T(m)=\text{exp}(-\int_{k_{n}}^{m}\sigma(\textbf{r}(m))dm)$ is an accumulated
transmittance along the ray. The parameters of $\Psi$ are optimized to produce
$\hat{C}(\textbf{r})$, which are expected to be close to the ground truth
pixel value.

StyleGAN Latent Code The expressiveness of the StyleGAN latent code $w$ can be
increased by inputting $k$ different latent codes, denoted by $\mathbf{w}$
($\mathbf{w}\in\mathcal{W}^{k}\subsetneq\mathbb{R}^{k\times 512}$)222For
simplicity, we use the notations ($\mathcal{W}^{k}$, $\mathcal{W}^{k}_{*}$)
from [47]., into each of the $k$ StyleGAN layers. This can further be extended
to $\mathcal{W}^{k}_{*}$ as introduced in [47]. Unlike $\mathcal{W}^{k}$,
whose distribution is bounded by the mapping network, $\mathcal{W}^{k}_{*}$
can reside outside the range of the mapping network. We choose to utilize
$\mathcal{W}^{k}_{*}$ in our pipeline to enable a wider range of attribute
manipulation.

EG3D EG3D [4] is a 3D-aware GAN based on NeRF. Feature vectors of a radiance
field are sampled from tri-plane generated by StyleGAN2 [24] and summed to a
tri-plane feature vector $F=g_{\theta}(\mathbf{x},\mathbf{w})$ given a 3D
position $\mathbf{x}\in\mathbb{R}^{3}$ and a latent code $\mathbf{w}$ that
modulates the StyleGAN2 layers, and $g_{\theta}$ denotes the function of the
whole process. $F$ is further processed by a small decoder to a
$M_{f}$-dimensional feature vector. Then a set of sampled features
$\\{{F_{ij}}\\}_{j=1}^{N_{s}}$ along the set of camera rays
$\\{\mathbf{r}_{i}\\}_{i=1}^{H_{V}\times W_{V}}$ is aggregated with volumetric
rendering using Eq. 1, where $N_{s}$ is the number of sampled points per ray,
and $H_{V}$ and $W_{V}$ are the height and the width of the rendered feature
image, respectively. Finally, the feature image is decoded into the high-
resolution image by a super-resolution module $\pi^{SR}$:

$\displaystyle\pi^{SR}:\mathbb{R}^{H_{V}\times W_{V}\times
M_{f}}\rightarrow\mathbb{R}^{H\times W\times 3}$ (2)

Figure 3: The overall architecture and the training objective of LeNeRF. (a)
The target latent code and the deformation field are generated by the latent
residual mapper (LRM) and the deformation network (DN), respectively. (b) The
tri-plane features for the source and target are encoded by the pretrained
generator and the soft mask for combining both features is estimated by our
attention field network (AFN). (c) By using the neural renderer (NR) which
incorporates the volume rendering and a super-resolution (not applied to
rendered masks), we obtain the rendered results for the source and target
features, mask, and the locally-edited features, which is the final output of
LeNeRF. (d) In the training process, only LRM, DN, and AFN are trained for
local editing, while other parameters for the generator and neural renderer
are fixed.

##  4 LeNeRF Framework for 3D Local Editing

In this section, we present Local Editing NeRF (LENeRF) which enables
localized editing of 3D scenes using a text prompt for manipulation,
$t_{edit}$ (e.g., blue eyes), and a prompt specifying the region of interest,
$t_{mask}$ (e.g., eyes). Unlike previous methods that perform global editing
by updating the latent codes of 3D GAN [48], we propose generating a 3D mask
using CLIP [39] supervision and performing localized editing with the
generated mask through a position-wise transformation of feature fields.

As shown in Fig. 3, LENeRF consists of frozen pretrained NeRF generator [4]
along with trainable add-on modules: Latent Residual Mapper (Section 4.1),
Attention Field Network (Section 4.2), and Deformation Network (Section 4.3).
The Latent Residual Mapper (LRM) maps the source latent code
$\mathbf{w}_{s}\in\mathcal{W}^{k}$ to the target latent code
$\mathbf{w}_{t}\in\mathcal{W}^{k}_{*}$, which in turn conditions the source
and target feature vectors $F_{s}$ and $F_{t}$ respectively (Fig. 3 (a)).

Then the 3D soft mask corresponding to the regions specified by $t_{mask}$ is
estimated by the Attention Field Network (AFN) (Fig. 3 (b)). The Deformation
Network handles geometric manipulations by deforming the source feature field
$\\{F_{s}\\}$, and the source and target features are fused to produce the
final feature fields. The modules are trained using the proposed CLIP loss
$\mathcal{L}_{CLIP^{+}}$, and the AFN is trained additionally with the mask
loss $\mathcal{L}_{mask}$ via pseudo-labels generated with relevance map
aggregation conditioned on the mask prompt $t_{mask}$ (Fig. 3 (d), Section
4.4). Once trained, LeNeRF performs 3D editing in real-time.

###  4.1 Latent Residual Mapper (LRM)

Inspired by CLIP-guided 2D image editing techniques [38, 12, 51, 49, 1], we
train the Latent Residual Mapper (Figure 3 (a)), a mapper function that
generates the target latent code $\mathbf{w}_{t}$ given the source latent code
$\mathbf{w}_{s}$. The mapper is trained using CLIP guidance, and once trained,
manipulation can be performed in real-time without the need for any inversion
steps [38].

Figure 4: Curated Examples of LENeRF.

Instead of using $\mathbf{w}_{t}$ directly for generating the final radiance
field, we utilize the mapper to produce tri-plane features $\\{F_{t}\\}$,
which are composited with the source tri-plane feature field $\\{F_{s}\\}$ to
generate the final feature field. This composition involves interpolating two
features based on an estimated 3D mask (Section 4.2).

The features in the 3D space are spatially entangled because the entire
feature field is conditioned with a single latent code. By utilizing more than
one latent code (in our case, two), we can grant position-wise freedom to the
features, which enables the localized editing of the source features based on
the 3D mask and the target features. Specifically, given the source latent
code $\mathbf{w}_{s}\in\mathcal{W}^{k}$, the mapper outputs the target latent
code $\mathbf{w}_{t}\in\mathcal{W}^{k}_{*}$ which is used to produce the final
feature field, described as:

$\displaystyle M(\mathbf{w}_{s})$
$\displaystyle=(M^{1}(\mathbf{w}_{s}^{1}),...,M^{N}(\mathbf{w}_{s}^{N}))$ (3)
$\displaystyle=(\Delta\mathbf{w}^{1},...,\Delta\mathbf{w}^{N}),$
$\displaystyle\mathbf{w}_{i}=(\mathbf{w}_{s}^{1}+\Delta\mathbf{w}^{1},...,\mathbf{w}_{s}^{N}+\Delta\mathbf{w}^{N}),$
(4)

where $M^{i},i\in{1,2,...,N}$ are fully-connected networks and
$\mathbf{w}^{n}$ denotes the subset of the latent code $\mathbf{w}$ that
modulates the $n$-th group of StyleGAN2-like backbone layers.

We design our mapper to estimate the residual values of latent codes, which
enables easier regularization and training. LRM is trained to minimize the
$\mathcal{L}_{CLIP^{+}}$ objective given a target prompt $t_{edit}$. See
Section 4.4 for more details.

###  4.2 Attention Field Network (AFN)

The Attention Field Network (AFN) parameterizes the 3D attention field that
serves as a soft 3D mask for interpolation between the source and the target
features. It is conditioned on both the source and target features, 3D
position, and the corresponding latent codes, allowing the model to take into
account both point semantics and global context:

$\displaystyle A:\mathbb{R}^{2M_{f}+6}\times\mathbb{R}^{2d}\rightarrow[0,1],$
(5) $\displaystyle
m=A(F_{s}\oplus\mathbf{x},F_{t}\oplus\mathbf{x},\mathbf{w}_{s},\mathbf{w}_{t}),$
$\displaystyle\hat{F}=(1-m)*F_{s}+m*F_{t},$ (6)

where $F_{s}$, $F_{t}\in\mathbb{R}^{M_{f}}$ are the source and target tri-
plane feature vectors, each conditioned by
$\mathbf{w}_{s},\mathbf{w}_{t}\in\mathbb{R}^{d}$ respectively, and
$m\in\mathbb{R}$ is the value of soft mask at $\mathbf{x}$.

For a position $\mathbf{x}$ inside the region of interest described by
$t_{mask}$, AFN is trained to produce a high mask value $m$ so that the
estimated feature $\hat{F}$ is dominated by the target feature $F_{t}$. For a
position $\mathbf{x}$ outside of our region of interest, low mask values are
estimated to ensure that the source feature remains unchanged. Utilizing an
estimated 3D soft mask enables 3D editing focused on desired regions, while
avoiding unintended changes in other areas. Note that we do not use any
domain-specific mask prediction model or semantic labels for AFN training.
Instead, we show that it is possible to distill CLIP’s capability to generate
zero-shot 2D masks into 3D using our proposed loss function, as explained in
Section 4.4.

###  4.3 Handling Large Deformations

Deformations are necessary for manipulations that involve large geometric
changes (e.g., opening mouth), as interpolating between features representing
different semantics (e.g., teeth and lips) often leads to artifacts (please
refer to the supplement for figures). The deformation field for the source
radiance field is estimated by the Deformation Network $T$ conditioned with
latent codes
($\mathbf{x}^{{}^{\prime}}=T(\mathbf{x},\mathbf{w}_{s},\mathbf{w}_{t})+\mathbf{x}$),
and Eq. 5 can be re-written as:

$\displaystyle
m=A(F_{s}\oplus\mathbf{x}^{{}^{\prime}},F_{t}\oplus\mathbf{x},\mathbf{w}_{s},\mathbf{w}_{t}).$
(7)

###  4.4 Training

Generator The estimated features (Eq. 7) are decoded into a radiance field and
rendered into a 2D image $\hat{I}_{t}$ via volume rendering (Eq. 1) followed
by the super-resolution module $\pi^{SR}$ (Eq. 2) given a randomly sampled
camera pose $\mathbf{v}\sim\mathcal{Z}_{v}$, where $\mathcal{Z}_{v}$ is the
camera pose distribution. Similarly, the source image $I_{s}$ and the raw
target image $I_{t}$ are rendered using the same process with $F_{s}$ and
$F_{t}$.

Figure 5: Results of partial editing.

CLIP Loss We use pretrained CLIP [39] text encoder $E_{T}$ and image encoder
$E_{I}$ to optimize our modules so that the estimated image $\hat{I}_{t}$
matches $t_{edit}$. However, a direct approach to maximize the cosine
similarity of $E_{I}(\hat{I}_{t})$ and $E_{T}(t_{edit}))$ frequently leads to
degenerate solutions. This is because the CLIP embedding space includes false
positives, where degenerate images result in a high similarity score with the
target text embedding. Therefore, we instead propose a robust
$\mathcal{L}_{CLIP}$ loss that utilizes contrastive loss together with image
augmentations:

$\displaystyle\mathcal{L}_{CLIP}=-\log{\frac{\sum_{s^{+}\in S^{+}}e^{q\cdot
s^{+}}}{\sum_{s\in S^{+}\cup S}e^{q\cdot s}}},$ (8)

where $S^{+}$ is a set of positive
samples333$S^{+}\vcentcolon=\\{E_{T}(t_{edit})-E_{I}(aug(I_{s})),E_{T}(t_{edit})-E_{T}(t_{src})\\}$
and $S^{-}$ is a set of negative
samples444$S^{-}\vcentcolon=\\{E_{T}(t_{edit})-E_{I}(aug(I_{s}))\\}$. Here,
$q=E_{I}(aug(\hat{I}_{t}))-E_{I}(aug(I_{s}))$, $t_{src}$ is a neutral text
describing the source image (e.g., ”Photo of a person”), and $aug$ refers to a
random image augmentation. $\mathcal{L}_{CLIP}$ maximizes the mutual
information between $q$ and the positive samples while minimizing that of the
negative samples. Comparing the directions of the CLIP embeddings and using
multiple cross-domain positive samples lead to stable and robust
optimizations, along with the negative samples inhibiting lazy manipulations.

Also, like [38], we incorporate an identity preserving loss [40]
$\mathcal{L}_{id}$, using a pretrained ArcFace [10] model, and a $L_{2}$ norm
of the manipulation step $M(\mathbf{w}_{s})$ to preserve the attributes of the
source radiance fields. The total loss for training the Latent Residual Mapper
and the Deformation Network is
$\mathcal{L}_{CLIP^{+}}={\mathcal{L}_{CLIP}+\lambda_{L2}\lVert
M_{(}\mathbf{w}_{s})\rVert}_{2}+\lambda_{id}\mathcal{L}_{id}$.

Figure 6: Results of out-of-distribution cases.

3D Distillation of CLIP’s Knowledge As we lack ground-truth 3D semantic masks
to supervise the 3D attention field, we render them into 2D and supervise them
using generated 2D pseudo-masks. This is necessary since training AFN directly
with $\mathcal{L}_{CLIP^{+}}$ often leads to the generation of irrelevant and
poorly localized attention fields, resulting in severe artifacts (Please see
supplement). This is not surprising, given that the target text prompt
$t_{edit}$ does not always align with the region of interest, and the CLIP
signal tends to have a global effect on the radiance fields. Furthermore, we
presume that additional complexity to the model without adequate
regularization easily leads to degenerate solutions.

Therefore we first introduce a separate masking prompt $t_{mask}$ solely
describing our region of interest. Next, we utilize $t_{mask}$ together with
the CLIP model to generate a 2D relevance mask of the source or target image
that serves as a pseudo-label for AFN training. Inspired by transformer
visualization methods [6, 2], we aggregate relevance maps across the
transformer heads and layers to build a relevance mask
$\mathbf{M}\in\mathbb{R}^{H_{V}\times W_{V}}$ with respect to the model output
$y$, given $t_{mask}$ (More details in the supplementary).

Unlike [6], where the transformer output $y_{t}$ is the logit value of class
$t$, our output $y$ is the cosine similarity between the embedding vector
$E_{t}(t_{mask})$ and $E_{I}(I)$. Generally, it is sufficient to use the
source image $I_{s}$ for the mask generation, but for large deformations, we
estimate two masks from both the source and intermediate images $I_{s}$ and
$I_{t}$ and perform a $\max$ operation.

The estimated relevance mask $\mathbf{M}$ is used as a pseudo-label to guide
the training of AFN by minimizing the MSE loss $\mathcal{L}_{mask}$ between
the volume-rendered attention field $\hat{\mathbf{M}}_{t}$ and $\mathbf{M}$.
While the generated pseudo-label $\mathbf{M}$ is low-dimensional and often
inaccurate, the semantic information contained in the input features to AFN
allows for training even with noisy guidance (Figure 3 (d)).

Figure 7: Results of LENeRF and various baselines.

Mask Regularization We utilize total variation regularization [41] (Eq. 9) to
regularize the 3D attention field for spatial smoothness, where
$\Delta_{x}^{2}m$ denotes the squared difference between $m_{i+1,j,k}$ and
$m_{i,j,k}$, and analogously for $\Delta_{y}^{2}m$ and $\Delta_{z}^{2}m$:

$\displaystyle\mathcal{L}_{tv}=\frac{1}{N}\sum_{m\in\mathcal{A}}{\sqrt{\Delta_{x}^{2}m+\Delta_{y}^{2}m+\Delta_{z}^{2}m}}.$
(9)

Also, the sparsity regularization (Eq. 10) is used in order to focus on the
region of interest:

$\displaystyle\mathcal{L}_{sparsity}=-\sum_{\mathbf{x}_{i}\in
K_{top}}{\log{m_{i}}}-\sum_{\mathbf{x}_{i}\in K_{bottom}}{log{(1-m_{i})}},$
(10)

where $K_{top}$ refers to the set of coordinates with the top-$k$ $m$ values,
and $K_{bottom}$ refers to the bottom-$k$. In summary, the objective function
for AFN $A$ is

$\displaystyle\mathcal{L}_{AFN}=\lambda_{mask}\mathcal{L}_{mask}+\lambda_{tv}\mathcal{L}_{tv}$
(11)
$\displaystyle+\lambda_{sparsity}\mathcal{L}_{sparsity}+\lambda_{CLIP^{+}}\mathcal{L}_{CLIP^{+}}.$

Please refer to the supplement for additional descriptions, hyperparameters,
and implementation details.

##  5 Experiments

Datasets LeNeRF utilizes pretrained EG3D [4] generator trained on FFHQ [23],
AFHQv2 CATS [9], and ShapeNet Cars [8, 28]. However, please note that the
datasets are not used when training LENeRF.

Baselines We compare LENeRF against three state-of-the-art NeRF editing
methods: CLIP-NeRF [48]555Note that the full code of CLIP-NeRF has not been
released at the time of writing. Therefore, the results are our reproduced
CLIP-NeRF., FENeRF [45], and IDE-3D [44]. While CLIP-NeRF is a text-guided
editing model, FENeRF and IDE-3D are not directly comparable to our method
since semantic masks are used as a control handle. Therefore we create two new
baselines based on the two models (FENeRF+StyleCLIP and IDE-3D+StyleCLIP)
which can edit radiance fields with text guidance. Specifically, we invert the
images generated by the two methods to StyleGAN latent codes using e4e [47]
encoder. StyleCLIP [38] is used to edit the inverted images given a target
text prompt. Face semantic maps are then extracted from the edited images
using a pretrained face parsing network of SofGAN [7], which is used for the
input guidance. Note that only FFHQ results can be reported for FENeRF and
IDE-3D since there are no available semantic labels and pretrained parsing
networks for other datasets.

Figure 8: Result of sequential editing.

Qualitative evaluations Fig. 4 provides curated results of our model, showing
the quality, diversity, and view consistency of LENeRF. Fig. 7 presents
qualitative comparisons against other three methods. FENeRF+StyleCLIP and
CLIP-NeRF fail to generate high-quality renderings, and the quality of images
degrades even further after editing. IDE-3D+StyleCLIP synthesizes high-
fidelity images before editing, but it is difficult to edit texture or
expression using semantic mask guidance. Also, semantic mask-based methods
require either inversion steps which cannot be done in real-time or an
encoding step which degrades identity preservation. LENeRF, however, can
perform localized editing in real-time and preserve the quality and the
identity of original content.

Fig. 5 shows partial editing results where we use the same attribute prompt
$t_{edit}$ while varying the mask prompt $t_{mask}$. This enables highly
localized editing without unwanted changes or color-bleeding artifacts. Fig. 8
demonstrates sequential editing results where we perform different
manipulations sequentially. Multiple changes to the source latent code destroy
the initial content gradually when trained without AFN (bottom) whereas, the
identity is preserved in LENeRF (top) by manipulating only the local parts of
the radiance field. Fig. 6 shows that LENeRF is robust to out-of-distribution
editing (e.g., fire on hair). LENeRF has the representation capability to
generate samples that are outside of the latent space by fusing the two
radiance fields, thereby reflecting the target prompts while preserving the
identity.

Table 1: PSNR, R-precision, and FID performance of various baselines and
LENeRF on FFHQ and Cats.

FFHQ Cats PSNR $\uparrow$ R-pre $\uparrow$ FID $\downarrow$ $\Delta$FID
$\downarrow$ FID $\downarrow$ $\Delta$FID $\downarrow$ CLIP-NeRF 7.08 0.21
41.5 +19.8 18.6 +12.7 FeNeRF + SC 5.33 0.19 37.4 +31.2 - - IDE3D + SC 12.44
0.61 4.56 +3.29 - - LeNeRF w/o AFN 10.81 0.66 $\mathbf{4.37}$ +4.90
$\mathbf{2.71}$ +4.17 LeNeRF (Ours) $\mathbf{20.86}$ $\mathbf{0.78}$
$\mathbf{4.37}$ +$\mathbf{2.21}$ $\mathbf{2.71}$ +$\mathbf{1.94}$

Figure 9: Results of the effect of $\lambda_{sparsity}$ on the mask.

Quantitative evaluations Table 1 provides quantitative measures of LENeRF and
various baselines. We quantify the unintended changes by computing the PSNR
between the source and the manipulated images outside of the region of
interest specified by $t_{mask}$. The ground-truth mask of the region is
detected by a pretrained face parsing network[7]. We assess the quality of the
source images and manipulated images using Fréchet Inception Distance (FID)
[18], and compare the differences to evaluate post-editing degradation. Also
for FFHQ, we report R-precision using a pretrained attribute classifier [30]
to measure how well the text prompt is reflected in the manipulated results.
We generate 30k samples with randomly sampled text prompts to calculate FID,
and sample 1K images per text prompt to calculate PSNR and R-precision.

Among the models, LENeRF shows the best FID score, demonstrating that it
produces high-quality results after the manipulation. Also, LENeRF have the
smallest FID difference between the images before and after the edit,
indicating the robustness of the model. Moreover, LENeRF shows the highest
PSNR and R-precision; it has the best performance in achieving the desired
manipulations while minimizing unintended changes. Please refer to the
supplement for more details.

User study We ask users to evaluate LENeRF along with various baselines in the
range from 1 to 10 regarding 1) fidelity, 2) locality, 3) identity
preservation, and 4) how well the text prompt is reflected in the results.
LENeRF outperforms all baselines by a large margin on each criterion, and the
scores are in the supplement.

Ablation study Fig. 9 shows the effect of controlling $\lambda_{sparsity}$. A
larger weight on $\mathcal{L}_{sparsity}$ produces a sharper 3D mask whereas a
smaller weight leads to a smoother mask. Fig. 10 visualizes the pixel-wise
difference map to show the effect of AFN and the radiance fusion technique on
the locality. Generating a 3D mask helps on minimize the unintended changes
outside of the region of interest and reduces color-bleeding artifacts that
are incorporated with the target prompt $t_{edit}$. Fig. 11 shows the
importance of the DN and our objective functions $\mathcal{L}_{CLIP^{+}}$ and
$\mathcal{L}_{mask}$. Minimizing the naive CLIP loss results in degenerate
solutions. Training AFN without $\mathcal{L}_{mask}$ fails to estimate a 3D
mask, resulting in severe artifacts. Finally, LENeRF without DN leads to the
interpolation of unrelated features and cannot handle geometric manipulations.

Figure 10: Results on comparison of pixel-wise difference maps. Figure 11:
Importance of our objective functions $\mathcal{L}_{CLIP^{+}}$,
$\mathcal{L}_{mask}$, and Deformation Network (DN).

##  6 Conclusion

Conclusion While texts provide rich and descriptive information for editing,
they often lack locality. LENeRF focuses on this aspect and proposes a point-
wise feature manipulation technique via 3D mask generation and radiance
fusion. We show that 3D-aware knowledge of NeRF and rich multi-modal
information from CLIP can be combined to create a robust 3D mask without any
additional datasets. LENeRF generates high-fidelity and find-grained 3D
editing results.

Acknowledgments. This work was supported by the Institute of Information &
communications Technology Planning & Evaluation (IITP) grant funded by the
Korean government (MSIT) (No. 2019-0-00075, Artificial Intelligence Graduate
School Program (KAIST) and No. 2021-0-01778, Development of human image
synthesis and discrimination technology below the perceptual threshold).

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# Algebraic Characterization of the Voronoi Cell Structure of the
$\mathcal{A}_{n}$ Lattice

Minho Kim

###### Abstract

We characterized the combinatorial structure of the Voronoi cell of the
$\mathcal{A}_{n}$ lattice in arbitrary dimensions. Based on the well-known
fact that the Voronoi cell is the disjoint union of $(n+1)!$ congruent
simplices, we show that it is the disjoint union of $(n+1)$ congruent hyper-
rhombi, which are the generalized rhombi or trigonal trapezohedra. The
explicit structure of the faces is investigated, including the fact that all
the $k$-dimensional faces, $2\leq k\leq n-1$, are hyper-rhombi. We show it to
be the vertex-first orthogonal projection of the $(n+1)$-dimensional unit
cube. Hence the Voronoi cell is a zonotope. We prove that in low dimensions
($n\leq 3$) the Voronoi cell can be understood as the section of that of the
$\mathcal{D}_{n+1}$ lattice with the hyperplane orthogonal to the diagonal
direction. We provide all the explicit coordinates and transformation matrices
associated with our analysis. Most of our analysis is algebraic and easily
accessible to those less familiar with the Coxeter-Dynkin diagrams.

##  1 Introduction

Due to their higher symmetry compared to other lattices, the root lattices
have been investigated in various fields including crystallography, signal
sampling, data discretization, etc. In many applications, the Voronoi cells of
root lattices play a crucial role.

Conway and Sloane [1] derived the Voronoi cells of root lattices (and their
dual lattices) based on the Coxeter-Dynkin diagrams. Specifically, they showed
that the Voronoi cell $V\left(\boldsymbol{0}\right)$ for a root lattice
$\mathcal{L}$ is the union of the images of its fundamental simplex under its
Weyl group $W_{\\!a}\\!\\!\left(\mathcal{L}\right)$. Moody and Patera [9]
determined the number of facets of all dimensions of the Voronoi cells of the
root lattices based on the the decorated Coxeter-Dynkin diagrams. Koca et al.
[7] analyzed the structure of the $(n-1)$-dimensional faces of the Voronoi
cells and related polytopes of the $\mathcal{A}_{n}$ and $\mathcal{D}_{n}$
lattices and their duals. They showed that the $(n-1)$-dimensional faces of
the Voronoi cells of the $\mathcal{A}_{n}$ lattice are the generalized
rhombohedra, which we call _hyper-rhombi_ in this work.

##  2 Notations

Our characterization uses the following notations.

  * •

$\boldsymbol{1}$ and $\boldsymbol{0}$ are the (column) vectors composed of
$1$s and $0$s whose dimensions are determined in the context.

  * •

$\boldsymbol{e}_{i}$ is the $i$-th standard unit (column) vector.

  * •

we denote the vectors with repeating elements as below [1].:

$(\underbrace{a,\dots,a}_{\alpha},\underbrace{b,\dots,b}_{\beta})=:(a^{\alpha},b^{\beta}).$

  * •

$\boldsymbol{I}_{n}$ is the identity and
$\boldsymbol{J}_{n}:=\boldsymbol{1}\boldsymbol{1}^{T}$.

  * •

$\boldsymbol{\mathit{\Sigma}}_{i,j}$ is the _row exchange_ matrix of the
$i$-th and $j$-th rows.

  * •

The $n\times n$ _circular shift_ matrix is defined as

$\boldsymbol{\mathit{\Gamma}}_{n,j}:=\begin{bmatrix}\boldsymbol{e}_{n-j+1}&\cdots&\boldsymbol{e}_{n}&\boldsymbol{e}_{1}&\cdots&\boldsymbol{e}_{n-j}\end{bmatrix}^{T}$
(1)

which is obtained by circularly shifting the rows of $\boldsymbol{I}_{n}$
downward by $j$.

  * •

The $n$-dimensional (closed) unit cube embedded in $\mathbb{R}^{n}$ is defined
as

$\square^{n}:=[0,1]^{n}=\left\\{\sum_{i=1}^{n}t_{i}\boldsymbol{e}_{i}:\boldsymbol{e}_{i}\in\mathbb{R}^{n},0\leq
t_{i}\leq 1\right\\}$

and

$\square^{n}_{\mathcal{J}}:=\left\\{\sum_{i\in\mathcal{J}}t_{i}\boldsymbol{e}_{i}:\boldsymbol{e}_{i}\in\mathbb{R}^{n},0\leq
t_{i}\leq 1\right\\},\quad\mathcal{J}\subset\\{1,\dots,n\\}$

is the $(n-\\#\mathcal{J})$-dimensional face of $\square^{n}$ adjacent to
$\boldsymbol{0}$ and spanned by the unit vectors excluding those associated
with the indices in $\mathcal{J}$.

  * •

The _convex hull_ of the point set
$\\{\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{k}\\}$ is denoted as

$\mathcal{CH}\left(\\{\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{k}\\}\right).$

##  3 Background

In this section we review the basic knowledge about the point groups
(represented by matrix groups), the root system $\mathscr{A}_{n}$, and
lattices.

###  3.1 Point Groups

A _point group_ is a group composed of _isometries_ that have a fixed point.
Here we represent a point group by a _matrix group_ with $\boldsymbol{0}$ as
the fixed point. Below are some point groups to be used in the main
discussion.

The _symmetric group_ $S_{n}$ is the finite group composed of $n!$ permutation
matrices. Below is a well-known triangulation of the unit cube based on the
symmetric group.

Figure 1:  Kuhn triangulation of the $3$-dimensional unit cube.

######  Lemma 1.

Given the $n$-dimensional simplex

$\tau_{0}:=\mathcal{CH}\left(\\{\boldsymbol{u}_{j}:0\leq j\leq
n\\}\right)\text{ where
}\boldsymbol{u}_{j}:=\boldsymbol{0}+\sum_{i=1}^{j}\boldsymbol{e}_{i}=(1^{j},0^{n-j}),$
(2)

the (non-trivial) disjoint union of the $n!$ images of $\tau_{0}$ under
$S_{n}$ forms the unit cube $\square^{n}$, i.e.,

$\square^{n}=\bigsqcup_{\mathclap{\boldsymbol{\mathit{\Pi}}\in
S_{n}}}\boldsymbol{\mathit{\Pi}}\tau_{0}.$

In addition, they all share the edge connecting $\boldsymbol{0}$ and
$\boldsymbol{1}$. Refer to Figure 1 for $n=3$.

###### Proof.

Refer to Kuhn [8]. This decomposition is called the _Kuhn triangulation_.
Since $\boldsymbol{0}$ and $\boldsymbol{1}$ are invariant under $S_{n}$, the
edge connecting them is shared by all the simplices. ∎

######  Corollary 1.

Let

$\boldsymbol{A}=\alpha\boldsymbol{I}_{n}+\beta\boldsymbol{J}_{n},\quad\alpha,\beta\in\mathbb{R}$

be invertible and $\boldsymbol{a}_{j}$ be the $j$-th column of
$\boldsymbol{A}$. The disjoint union of the $n!$ images of the $n$-simplex

$\mathcal{CH}\left(\left\\{\boldsymbol{0}+\sum_{i=1}^{j}\boldsymbol{a}_{i}:0\leq
j\leq n\right\\}\right)$

under $S_{n}$ forms an $n$-dimensional parallelepiped
$\boldsymbol{A}\square^{n}$ and the simplices share the edge connecting
$\boldsymbol{0}$ and $\sum_{j=1}^{n}\boldsymbol{a}_{j}$.

###### Proof.

The claim holds since, for $\boldsymbol{\mathit{\Pi}}\in S_{n}$,

$\boldsymbol{\mathit{\Pi}}\boldsymbol{A}=\boldsymbol{A}\boldsymbol{\mathit{\Pi}}$

and therefore

$\boldsymbol{\mathit{\Pi}}\left(\sum_{j\in\mathcal{J}}\boldsymbol{a}_{j}\right)=\boldsymbol{\mathit{\Pi}}\left(\sum_{j\in\mathcal{J}}\boldsymbol{A}\boldsymbol{e}_{j}\right)=\boldsymbol{A}\left(\boldsymbol{\mathit{\Pi}}\sum_{j\in\mathcal{J}}\boldsymbol{e}_{j}\right),\quad\mathcal{J}\subset\\{1,\dots,n\\}.$

∎

The _circular shift group_

$C_{n}:=\\{\boldsymbol{\mathit{\Gamma}}_{n,i}:0\leq i\leq n-1\\},$

is the finite group composed of $n$ circular shift matrices (1). Given a set
of the hyperplanes through $\boldsymbol{0}$, A _reflection group_ is a
discrete group generated by the reflections w.r.t. the given hyperplanes
through $\boldsymbol{0}$ [4].

###  3.2 Root System $\mathscr{A}_{n}$

The _root system_ $\mathscr{A}_{n}$ is composed of the following $n(n+1)$
_roots_

$\mathscr{A}_{n}=\\{\boldsymbol{e}_{i}-\boldsymbol{e}_{j}\in\mathbb{R}^{n+1}:1\leq
i\neq j\leq n+1\\}$

embedded in the $n$-dimensional hyperplane

$\mathbb{H}_{\boldsymbol{1}}:=\left\\{\boldsymbol{x}\in\mathbb{R}^{n+1}:\boldsymbol{x}\cdot\boldsymbol{1}=0\right\\}.$
(3)

The reflections w.r.t. the hyperplanes orthogonal to the roots in
$\mathscr{A}_{n}$ form a finite reflection group called the _Weyl group_
$W\\!\\!\left(\mathscr{A}_{n}\right)$.

###  3.3 Lattices

Given an $m\times n$ generator matrix $\boldsymbol{G}$ with $m\geq n$ and
$\operatorname{rank}(\boldsymbol{G})=n$, the _lattice_ $\mathcal{L}_{n}$
embedded in $\mathbb{R}^{m}$ is defined as all the integer linear combinations
of the columns of $\boldsymbol{G}$:

$\mathcal{L}_{n}:=\boldsymbol{G}\mathbb{Z}^{n}=\\{\boldsymbol{G}\boldsymbol{j}\in\mathbb{R}^{m}:\boldsymbol{j}\in\mathbb{Z}^{n}\\}.$

The _dual lattice_ of $\mathcal{L}_{n}$ is defined as

$\mathcal{L}_{n}^{*}:=\\{\boldsymbol{x}\in\mathbb{R}^{m}:\boldsymbol{x}\cdot\boldsymbol{y}\in\mathbb{Z},\forall\boldsymbol{y}\in\mathcal{L}_{n}\\}.$

The _Voronoi cell_ at $\boldsymbol{p}\in\mathcal{L}_{n}$ is defined as the set
of points closest to $\boldsymbol{p}$ among all the lattice points of
$\mathcal{L}_{n}$

$V_{\mathcal{L}_{n}}\left(\boldsymbol{p}\right):=\\{\boldsymbol{x}\in\mathbb{R}^{m}:\|\boldsymbol{x}-\boldsymbol{p}\|\leq\|\boldsymbol{x}-\boldsymbol{q}\|,\text{
for all }\boldsymbol{q}\in\mathcal{L}_{n}\\}.$

##  4 The Root Lattice $\mathcal{A}_{n}$

(a) $W\\!\\!\left(\mathscr{A}_{2}\right)$

(b) $W_{\\!a}\\!\\!\left(\mathscr{A}_{2}\right)$

(c) $V\left(\boldsymbol{0}\right)$

Figure 2:  2(a) $W\\!\\!\left(\mathscr{A}_{2}\right)$ with the fundamental
region (gray area), 2(b) $W_{\\!a}\\!\\!\left(\mathscr{A}_{2}\right)$ with the
fundamental simplex (gray triangle), and 2(c) $V\left(\boldsymbol{0}\right)$
(thick black hexagon) decomposed into three rhombi, ${\Diamond}_{1}$ (red),
${\Diamond}_{2}$ (green), and ${\Diamond}_{3}$ (blue).

(a) $W\\!\\!\left(\mathscr{A}_{3}\right)$

(b) $W_{\\!a}\\!\\!\left(\mathscr{A}_{3}\right)$

Figure 3:  3(a) The six hyperplanes (denoted by the great circles) associated
with the twelve roots of $\mathscr{A}_{3}$, projected onto $\mathbb{R}^{3}$ by
$\boldsymbol{P}_{3}$. The three colored hyperplanes form the fundamental
region whose intersection with the sphere is colored in gray. 3(b) The Voronoi
cell (rhombic dodecahedron) of the $\mathcal{A}_{3}$ lattice with the
fundamental simplex of $\mathscr{A}_{3}$ (gray tetrahedron).

In this section, we review the basic properties of the $\mathcal{A}_{n}$
lattice based on an explicit representation given by a generator matrix.

###  4.1 Definition and Basic Properties

The $\mathcal{A}_{n}$ lattice is composed of all the integer linear
combinations of the roots of $\mathscr{A}_{n}$ and usually defined as

$\mathbb{Z}_{\\!\\!\mathcal{A}}^{n}:=\boldsymbol{G}_{\\!\\!\mathcal{A}}\mathbb{Z}^{n}=\left\\{\boldsymbol{x}\in\mathbb{Z}^{n+1}:\boldsymbol{x}\cdot\boldsymbol{1}=0\right\\}=\mathbb{Z}^{n+1}\cap\mathbb{H}_{\boldsymbol{1}}$

embedded in $\mathbb{H}_{\boldsymbol{1}}$ (3) and generated by the
$(n+1)\times n$ generator matrix [3]

$\boldsymbol{G}_{\\!\\!\mathcal{A}}:=\begin{bmatrix}[r]1\\\ -1&1\\\
&-1&\ddots\\\ &&\ddots&1\\\ &&&-1&1\\\ &&&&-1\\\ \end{bmatrix}.$

The $n$ hyperplanes orthogonal to them enclose the _fundamental chamber_ or
_fundamental region_ (Figure 2(a) and Figure 3(a)) and its images under the
_Weyl group_ $W\\!\\!\left(\mathscr{A}_{n}\right)$ fill
$\mathbb{H}_{\boldsymbol{1}}$.

Let

$\boldsymbol{\Xi}:=\frac{1}{n+1}\begin{bmatrix}[r]n&-1&\dots&-1&-1\\\
-1&n&\dots&-1&-1\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\ -1&-1&\dots&n&-1\\\
-1&-1&\dots&-1&n\\\
\end{bmatrix}=\boldsymbol{I}_{n+1}-\frac{1}{n+1}\boldsymbol{J}_{n+1}$

be the orthogonal projection matrix onto $\mathbb{H}_{\boldsymbol{1}}$ along
$\boldsymbol{1}$ and $\boldsymbol{\xi}_{k}$ be the $k$-th column of
$\boldsymbol{\Xi}$. The dual lattice of $\mathbb{Z}_{\\!\\!\mathcal{A}}^{n}$
is defined as
$\mathbb{Z}_{\\!\\!\mathcal{A}^{*}}^{n}:=\boldsymbol{G}_{\\!\\!\mathcal{A}^{*}}\mathbb{Z}^{n}$
where [3]

$\boldsymbol{G}_{\\!\\!\mathcal{A}^{*}}:=\begin{bmatrix}\boldsymbol{\xi}_{1}&\cdots&\boldsymbol{\xi}_{n}\end{bmatrix}.$

We can construct both $\mathcal{A}_{n}$ and $\mathcal{A}_{n}^{*}$ lattices in
$\mathbb{R}^{n}$ using any $n\times(n+1)$ orthogonal transformation that maps
$\mathbb{H}_{\boldsymbol{1}}$ to $\mathbb{R}^{n}$ and one of such matrices is
[6]

$\boldsymbol{P}_{n}:=\left(\boldsymbol{I}_{n}-\frac{1}{n}\left(1+\frac{1}{\sqrt{n+1}}\right)\boldsymbol{J}_{n}\right)\begin{bmatrix}-\boldsymbol{I}_{n}&\boldsymbol{1}\end{bmatrix}.$

For example, with

$\boldsymbol{P}_{3}=\frac{1}{2}\begin{bmatrix}[r]-1&1&1&-1\\\ 1&-1&1&-1\\\
1&1&-1&-1\end{bmatrix}$

we get the following square generator matrices for the $\mathcal{A}_{3}$ and
$\mathcal{A}_{3}^{*}$ lattices which are equivalent to the FCC (Face-Centered
Cubic) and BCC (Body-Centered Cubic) lattices, respectively.:

$\boldsymbol{P}_{3}\boldsymbol{G}_{\\!\\!\mathcal{A}}=\begin{bmatrix}[r]-1&0&1\\\
1&-1&1\\\ 0&1&0\end{bmatrix}\text{ and
}\boldsymbol{P}_{3}\boldsymbol{G}_{\\!\\!\mathcal{A}^{*}}=\frac{1}{2}\begin{bmatrix}[r]-1&1&1\\\
1&-1&1\\\ 1&1&-1\end{bmatrix}.$

Note that

$\left(\boldsymbol{P}_{3}\boldsymbol{G}_{\\!\\!\mathcal{A}}\right)\mathbb{Z}^{3}=\boldsymbol{G}_{\mathrm{fcc}}\mathbb{Z}^{3}\text{
and
}\boldsymbol{P}_{3}\boldsymbol{G}_{\\!\\!\mathcal{A}^{*}}=\boldsymbol{G}_{\mathrm{bcc}}$

where

$\boldsymbol{G}_{\mathrm{fcc}}:=\begin{bmatrix}0&1&1\\\ 1&0&1\\\ 1&1&0\\\
\end{bmatrix}\text{ and
}\boldsymbol{G}_{\mathrm{bcc}}:=\frac{1}{2}\begin{bmatrix}[r]-1&1&1\\\
1&-1&1\\\ 1&1&-1\\\ \end{bmatrix}$

are the well-known symmetric generator matrices for the FCC and BCC lattices,
respectively.

We review a well-known fact of the $\mathcal{A}_{n}$ lattice.

######  Lemma 2.

The reflection group generated by the reflections w.r.t. the hyperplanes
orthogonal to the roots of $\mathscr{A}_{n}$ is isomorphic to the symmetric
group $S_{n+1}$.

###### Proof.

Let

$\displaystyle\boldsymbol{y}:=\boldsymbol{x}-2\frac{\boldsymbol{x}\cdot(\boldsymbol{e}_{i}-\boldsymbol{e}_{j})}{\|\boldsymbol{e}_{i}-\boldsymbol{e}_{j}\|^{2}}(\boldsymbol{e}_{i}-\boldsymbol{e}_{j})=\boldsymbol{x}-(\boldsymbol{x}(i)-\boldsymbol{x}(j))(\boldsymbol{e}_{i}-\boldsymbol{e}_{j})$

be the reflected image of $\boldsymbol{x}$ w.r.t. the hyperplane orthogonal to
$\boldsymbol{e}_{i}-\boldsymbol{e}_{j}$. Then

$\boldsymbol{y}(k)=\begin{cases}\boldsymbol{x}(i)-\boldsymbol{x}(i)+\boldsymbol{x}(j)=\boldsymbol{x}(j)&\text{if
}k=i\\\
\boldsymbol{x}(j)+\boldsymbol{x}(i)-\boldsymbol{x}(j)=\boldsymbol{x}(i)&\text{if
}k=j\\\ \boldsymbol{x}(k)&\text{otherwise}.\end{cases}$

and therefore
$\boldsymbol{y}=\boldsymbol{\mathit{\Sigma}}_{i,j}\boldsymbol{x}$. The claim
holds since $S_{n+1}$ can be generated by row-exchange matrices. ∎

##  5 Decomposition of the Voronoi Cell into Congruent Simplices

In this and the following sections we algebraically analyze the detailed
structure of the Voronoi cell of the $\mathcal{A}_{n}$ lattice. Since all the
Voronoi cells are congruent, we only consider $V\left(\boldsymbol{0}\right)$
at the origin.

If we add the hyperplane
$(\boldsymbol{e}_{1}-\boldsymbol{e}_{n+1})\cdot\boldsymbol{x}=1$ to the
hyperplanes (orthogonal to the roots) of $\mathscr{A}_{n}$, we obtain an
infinite reflection group called the _affine Weyl group_
$W_{\\!a}\\!\\!\left(\mathscr{A}_{n}\right)$ (Figure 2(a)). The simplex built
by ‘capping’ the fundamental region of $\mathscr{A}_{n}$ with the additional
hyperplane is called the _fundamental simplex_ (or _alcove_) of
$\mathscr{A}_{n}$ [3].

We start with the following lemma by Conway and Sloane [1] that shows
$V\left(\boldsymbol{0}\right)$ is composed of $(n+1)!$ copies of the
fundamental simplex of $W_{\\!a}\\!\\!\left(\mathscr{A}_{n}\right)$.

######  Lemma 3.

Let (Refer to Lemma 1 for $\boldsymbol{u}_{j}$.)

$\mathcal{V}_{0}:=\left\\{\boldsymbol{v}_{j}:=\boldsymbol{\Xi}\boldsymbol{u}_{j}=\boldsymbol{0}+\sum_{i=1}^{j}\boldsymbol{\xi}_{i}=\left(\left(\frac{n+1-j}{n+1}\right)^{j},\left(-\frac{j}{n+1}\right)^{n+1-j}\right):0\leq
j\leq n\right\\}.$ (4)

$V\left(\boldsymbol{0}\right)$ is the union of the $(n+1)!$ images of the
fundamental simplex

$\sigma_{0}:=\mathcal{CH}\left(\mathcal{V}_{0}\right)$ (5)

under the symmetric group $S_{n+1}$. In other words,

$V\left(\boldsymbol{0}\right)=\bigsqcup_{\mathclap{\boldsymbol{\mathit{\Pi}}\in
S_{n+1}}}\boldsymbol{\mathit{\Pi}}\sigma_{0}.$

The simplicial face
$\mathcal{CH}\left(\\{\boldsymbol{v}_{1},\dots,\boldsymbol{v}_{n}\\}\right)$
is called the _roof_ of $\sigma_{0}$.

Refer to Figure 2(b) and Figure 3(b) for $n=2$ and $n=3$, respectively.

###### Proof.

Refer to Conway and Sloane [1, III.B]. ∎

Therefore, the vertices of $V\left(\boldsymbol{0}\right)$ is the union of
$(n+1)!$ images of $\mathcal{V}_{0}$ excluding $\boldsymbol{0}$, i.e.,

$\bigcup_{i=1}^{n}\left\\{\sum_{\boldsymbol{\zeta}\in\boldsymbol{\mathrm{Z}}}\boldsymbol{\zeta}:\boldsymbol{\mathrm{Z}}\subset\boldsymbol{\Xi},\\#\boldsymbol{\mathrm{Z}}=i\right\\}$

and the number of vertices of $V\left(\boldsymbol{0}\right)$ is [2]

$\sum_{i=1}^{n}\binom{n+1}{i}=2^{n+1}-2.$

For example, (Note that
$\sum_{i=1}^{n+1}\boldsymbol{\xi}_{i}=\boldsymbol{0}$.) for $n=2$, the six
vertices are (Figure 2(c))

$\left\\{\pm\boldsymbol{\xi}_{1},\pm\boldsymbol{\xi}_{2},\pm\boldsymbol{\xi}_{3}\right\\}$

and for $n=3$, the $14$ vertices are (Figure 3(b))

$\left\\{\pm\boldsymbol{\xi}_{1},\pm\boldsymbol{\xi}_{2},\pm\boldsymbol{\xi}_{3},\pm\boldsymbol{\xi}_{4},\pm(\boldsymbol{\xi}_{1}+\boldsymbol{\xi}_{2}),\pm(\boldsymbol{\xi}_{1}+\boldsymbol{\xi}_{3}),\pm(\boldsymbol{\xi}_{1}+\boldsymbol{\xi}_{4})\right\\}.$

##  6 Decomposition of the Voronoi Cell into Congruent Hyper-Rhombi

In this section we show that $V\left(\boldsymbol{0}\right)$ is composed of
$(n+1)$ congruent ‘hyper-rhombi’, a generalization of rhombi or _trigonal
trapezohedra_ (_rhombohedra_ with six congruent rhombic faces).

######  Definition 1.

An $n$-dimensional ‘hyper-rhombus’ is defined as a parallelepiped all of whose
faces are congruent $(n-1)$-dimensional hyper-rhombi. This definition is
recursive with its base case of $2$-dimensional rhombi (Figure 4).

Let

$\boldsymbol{\Xi}_{\mathcal{J}}:=\boldsymbol{\Xi}\backslash\\{\boldsymbol{\xi}_{j}:j\in\mathcal{J}\\}$

be the set of column vectors of $\boldsymbol{\Xi}$ excluding those associated
with the indices in $\mathcal{J}$. To avoid notational clutter, we denote
$\boldsymbol{\Xi}_{\\{i\\}}=\boldsymbol{\Xi}_{i}$ and
$\boldsymbol{\Xi}_{\\{i,j\\}}=\boldsymbol{\Xi}_{ij}$, etc. Then, for
$\mathcal{J}\neq\emptyset$,

${\Diamond}_{\mathcal{J}}:=\boldsymbol{\Xi}_{\mathcal{J}}[0,1]^{n+1-\\#\mathcal{J}}=\left\\{\sum_{\mathclap{\begin{subarray}{c}i=1\\\
i\notin\mathcal{J}\end{subarray}}}^{n+1}t_{i}\boldsymbol{\xi}_{i}:0\leq
t_{i}\leq 1\right\\}$

is the $(n+1-\\#\mathcal{J})$-dimensional parallelepiped spanned by the
directions in $\boldsymbol{\Xi}_{\mathcal{J}}$ since any $n$ directions of
$\boldsymbol{\Xi}$ are linearly independent.

######  Lemma 4.

The $n$-dimensional parallelepipeds $\left\\{{\Diamond}_{j}:1\leq j\leq
n+1\right\\}$ are hyper-rhombi.

###### Proof.

Note that the $n$ faces of ${\Diamond}_{j}$ adjacent to $\boldsymbol{0}$ are

$\left\\{{\Diamond}_{ij}:1\leq i\neq j\leq n+1\right\\}.$

They are all congruent with ${\Diamond}_{1,2}$ since (with
$\boldsymbol{\Xi}_{ij}$ viewed as a multi-set)

$\boldsymbol{\Xi}_{ij}=\boldsymbol{\mathit{\Sigma}}_{1,i}\boldsymbol{\mathit{\Sigma}}_{2,j}\boldsymbol{\Xi}_{1,2}$

and therefore are all congruent. Similarly, all the $(n-2)$-dimensional faces
of ${\Diamond}_{ij}$ adjacent to $\boldsymbol{0}$,

$\left\\{{\Diamond}_{ijk}:1\leq k\neq i,j\leq n+1\right\\},$

are also congruent parallelepipeds. Finally, all the $2$-dimensional faces
spanned by two columns of $\boldsymbol{\Xi}$ are rhombi since

$\boldsymbol{\xi}_{\alpha}\cdot\boldsymbol{\xi}_{\beta}=\begin{cases}n/(n+1)&\alpha=\beta\\\
-1/(n+1)&\alpha\neq\beta\end{cases}$

and therefore the claim holds. ∎

From the above lemma, we can deduce the following Corollary.

######  Corollary 2.

For any $\boldsymbol{\Xi}_{\mathcal{J}}$ where
$\mathcal{J}\subset\\{1,\dots,n+1\\}$ and $2\leq\\#\mathcal{J}\leq n$,
${\Diamond}_{\mathcal{J}}$ is a hyper-rhombus.

Next we show that $V\left(\boldsymbol{0}\right)$ is the disjoint union of
$(n+1)$ congruent hyper-rhombi.

Figure 4:  4 Decomposition of $V\left(\boldsymbol{0}\right)$ of
$\mathbb{Z}_{\\!\\!\mathcal{A}}^{3}$ into four hyper-rhombi: (green)
${\Diamond}_{1}$, (red) ${\Diamond}_{2}$, (gray) ${\Diamond}_{3}$, and (blue)
${\Diamond}_{4}$. 4 ${\Diamond}_{4}$ with its vertices. 4 Decomposition of
${\Diamond}_{4}$ into six tetrahedra. 4 The other hyper-rhombic decomposition.

######  Lemma 5.

$V\left(\boldsymbol{0}\right)$ is the disjoint union of $(n+1)$ congruent
hyper-rhombi and each hyper-rhombus is further composed of $n!$ congruent
simplices (Figure 2(c) and Figure 4). Specifically, the union of the $n!$
images of $\sigma_{0}$ (5) under the permutations of the first $n$ components
form the $n$-dimensional hyper-rhombus ${\Diamond}_{n+1}$ (the blue rhombus in
Figure 2(c) and the blue rhombohedron in Figure 4). Then, the union of the
$(n+1)$ images of ${\Diamond}_{n+1}$ under the circular shifts in $C_{n+1}$
forms $V\left(\boldsymbol{0}\right)$. In other words,

$V\left(\boldsymbol{0}\right)=\bigsqcup_{\mathclap{\boldsymbol{\mathit{\Gamma}}\in
C_{n+1}}}\boldsymbol{\mathit{\Gamma}}{\Diamond}_{n+1}=\bigsqcup_{j=1}^{n+1}{\Diamond}_{j}\text{
where
}{\Diamond}_{j}=\boldsymbol{\mathit{\Gamma}}_{n+1,j\bmod(n+1)}{\Diamond}_{n+1}$

and

${\Diamond}_{n+1}=\bigsqcup_{\mathclap{\boldsymbol{Q}\in
G_{1}}}\boldsymbol{Q}\sigma_{0}\text{ where
}G_{1}:=\left\\{\begin{bmatrix}\boldsymbol{\mathit{\Pi}}\\\ &1\\\
\end{bmatrix}:\boldsymbol{\mathit{\Pi}}\in S_{n}\right\\}.$

###### Proof.

We already know that ${\Diamond}_{n+1}$ is a hyper-rhombus due to Lemma 4.
Since the vertices of $\mathcal{V}_{0}$ are taken from

$\boldsymbol{\Xi}_{n+1}=\begin{bmatrix}\boldsymbol{\xi}_{1}&\cdots&\boldsymbol{\xi}_{n}\end{bmatrix}=\begin{bmatrix}\boldsymbol{I}_{n}+\left(-\frac{1}{n+1}\right)\boldsymbol{J}_{n}\\\
\boldsymbol{1}^{T}\end{bmatrix}$

and due to Corollary 1, the union of the images of $\sigma_{0}$ under $G_{1}$
forms an $n$-dimensional hyper-rhombus embedded in the hyperplane
$\boldsymbol{x}\cdot\boldsymbol{e}_{n+1}=1$. Then, the $(n+1)$ parallelepipeds
$\left\\{{\Diamond}_{j}:1\leq j\leq n+1\right\\}$ are closed under the actions
in $C_{n+1}$ since (with $\boldsymbol{\Xi}_{n+1}$ and $\boldsymbol{\Xi}_{j}$
viewed as sets)

$\boldsymbol{\mathit{\Gamma}}_{n+1,j\bmod(n+1)}\boldsymbol{\Xi}_{n+1}=\boldsymbol{\Xi}_{j},\quad
1\leq j\leq n+1.$

Finally, the union of them forms $V\left(\boldsymbol{0}\right)$ due to Lemma 3
and the decomposition

$S_{n+1}=C_{n+1}G_{1}.$

∎

Note that there is another hyper-rhombic decomposition

$V\left(\boldsymbol{0}\right)=\bigsqcup_{j=1}^{n+1}\left(\boldsymbol{\xi}_{j}+{\Diamond}_{j}\right)$
(6)

which can be obtained by ‘flipping’ all the directions in $\boldsymbol{\Xi}$
(Figure 4).

######  Corollary 3.

We can compute the volume of the Voronoi cell as

$\operatorname{vol}\left(V\left(\boldsymbol{0}\right)\right)=(n+1)\operatorname{vol}\left({\Diamond}_{n+1}\right)=(n+1)\sqrt{\boldsymbol{\Xi}_{n+1}^{T}\boldsymbol{\Xi}_{n+1}}=\sqrt{n+1}$

which was verified by other researchers [1, 7].

######  Lemma 6.

All the $k$-dimensional faces of $V\left(\boldsymbol{0}\right)$, $2\leq k\leq
n-1$, are hyper-rhombi. See Figure 6 for $n=4$.

###### Proof.

The claim holds due to Corollary 2 and Lemma 5. ∎

Note that Koca et al. [7] proved the case of $k=n-1$.

Figure 5:  5 (Blue triangle) the roof of the fundamental simplex. 5 (Blue
rhombus) the hyper-rhombic face $\boldsymbol{\xi}_{1}+{\Diamond}_{14}$. The
triangle
$\mathcal{CH}\left(\boldsymbol{\xi}_{1},\boldsymbol{\xi}_{1}+\boldsymbol{\xi}_{3},-\boldsymbol{\xi}_{4}\right)$
is obtained by applying $\begin{bmatrix}1\\\ &&1\\\ &1\\\ &&&1\end{bmatrix}$
to 5. 5 Three hyper-rhombic faces sharing $-\boldsymbol{\xi}_{4}$: (blue)
$\boldsymbol{\xi}_{1}+{\Diamond}_{14}$, (green)
$\boldsymbol{\xi}_{2}+{\Diamond}_{24}$, and (red)
$\boldsymbol{\xi}_{3}+{\Diamond}_{34}$. The red and green hyper-rhombic faces
are obtained by applying $\begin{bmatrix}&1\\\ &&1\\\ 1\\\ &&&1\end{bmatrix}$
and $\begin{bmatrix}&&1\\\ 1\\\ &1\\\ &&&1\end{bmatrix}$ to the blue face,
respectively. The colored arrows denote the nearest lattice points of
$V\left(\boldsymbol{0}\right)$ associated with each face. The whole $12$ faces
are obtained by applying the transformations in $C_{4}$ to 5.  Figure 6:  The
hyper-rhombic face $\boldsymbol{\xi}_{1}+{\Diamond}_{15}$ of
$V\left(\boldsymbol{0}\right)$ of $\mathbb{Z}_{\\!\\!\mathcal{A}}^{4}$. The
tetrahedron
$\mathcal{CH}\left(\left\\{\sum_{i=1}^{j}\boldsymbol{\xi}_{i}:1\leq j\leq
4\right\\}\right)$ is the roof of $\sigma_{0}$.

Next, we analyze the structure of the _cells_ , $(n-1)$-dimensional faces, of
$V\left(\boldsymbol{0}\right)$. We first determine the explicit structure of
the hyper-rhombic face that contains the roof of the fundamental simplex.

######  Lemma 7.

The hyper-rhombic cell of $V\left(\boldsymbol{0}\right)$ that contains the
roof of $\sigma_{0}$ is

$\boldsymbol{\xi}_{1}+{\Diamond}_{1,n+1}$

which is composed of the $(n-1)!$ images of the roof of $\sigma_{0}$ under the
group

$G_{2}:=\left\\{\begin{bmatrix}1\\\ &\boldsymbol{\mathit{\Pi}}\\\
&&1\end{bmatrix}:\boldsymbol{\mathit{\Pi}}\in S_{n-1}\right\\}$ (7)

and they all share the edge connecting $\boldsymbol{\xi}_{1}$ and
$-\boldsymbol{\xi}_{n+1}$. Refer to Figure 5 and Figure 6 for $n=3$ and $n=4$,
respectively.

###### Proof.

Due to Lemma 3, the roof of $\sigma_{0}$ is

$\mathcal{CH}\left(\left\\{\boldsymbol{v}_{j}=\boldsymbol{\xi}_{1}+\sum_{i=2}^{j}:1\leq
j\leq n\right\\}\right)$

where the vertices are taken from the $(n+1)\times(n-1)$ matrix

$\boldsymbol{\Xi}_{1,n+1}=\begin{bmatrix}\\\
\boldsymbol{\xi}_{2}&\cdots&\boldsymbol{\xi}_{n}\\\ \\\
\end{bmatrix}=\begin{bmatrix}\boldsymbol{1}^{T}\\\
\boldsymbol{I}_{n-1}+\left(-\frac{1}{n+1}\right)\boldsymbol{J}_{n-1}\\\
\boldsymbol{1}^{T}\\\ \end{bmatrix}.$

Due to Corollary 1, the union of the images of
$\mathcal{CH}\left(\left\\{\boldsymbol{v}_{j}:2\leq j\leq n\right\\}\right)$
under $G_{2}$ forms the $(n-1)$-dimensional hyper-rhombus ${\Diamond}_{1,n+1}$
embedded in the $(n-1)$-dimensional hyperplane

$\left\\{\boldsymbol{x}\cdot\boldsymbol{e}_{1}=1:\boldsymbol{x}\in\mathbb{R}^{n+1}\right\\}\cap\left\\{\boldsymbol{x}\cdot\boldsymbol{e}_{n+1}=1:\boldsymbol{x}\in\mathbb{R}^{n+1}\right\\}$

and they all share the edge connecting $\boldsymbol{0}$ and
$\sum_{i=2}^{n}\boldsymbol{\xi}_{i}$. Moreover, since $\boldsymbol{\xi}_{1}$
is invariant under $G_{2}$ the ‘shifted’ simplices all share the edge
connecting $\boldsymbol{\xi}_{1}$ and
$\boldsymbol{\xi}_{1}+\sum_{i=2}^{n}\boldsymbol{\xi}_{i}=-\boldsymbol{\xi}_{n+1}$.
∎

Next we show that the vertex $-\boldsymbol{\xi}_{n+1}$ of
$V\left(\boldsymbol{0}\right)$ is surrounded by $n$ non-overlapping congruent
hyper-rhombic cells.

######  Lemma 8.

On the boundary of $V\left(\boldsymbol{0}\right)$, there are $n$ congruent
hyper-rhombic cells sharing the vertex $-\boldsymbol{\xi}_{n+1}$. These cells
are

  1. (i)

the images of $\boldsymbol{\xi}_{1}+{\Diamond}_{1,n+1}$ under the group of the
circular shifts of the first $n$ components and

  2. (ii)

the $(n-1)$-dimensional hyper-rhombic cells of ${\Diamond}_{n+1}$ adjacent to
$-\boldsymbol{\xi}_{n+1}$.

Refer to Figure 2(c) and Figure 5.

###### Proof.

Let

$G_{3}:=\left\\{\begin{bmatrix}\boldsymbol{\mathit{\Gamma}}_{n,i}\\\
&1\end{bmatrix}:\boldsymbol{\mathit{\Gamma}}_{n,i}\in C_{n},0\leq i\leq
n-1\right\\}.$ (8)

Since $G_{2},G_{3}\leq S_{n+1}$, $G_{2}\cap G_{3}=\\{\boldsymbol{I}_{n+1}\\}$,
and all the cells of $V\left(\boldsymbol{0}\right)$ are composed of the images
of the roof of $\sigma_{0}$ under $S_{n+1}$, the $n$ images of
$\boldsymbol{\xi}_{1}+{\Diamond}_{1,n+1}$ under $G_{3}$ do not overlap.

Moreover, since ($0\leq i\leq n$)

$\begin{bmatrix}\boldsymbol{\mathit{\Gamma}}_{n,i}\\\
&1\end{bmatrix}(\boldsymbol{\xi}_{1}+{\Diamond}_{1,n+1})=\boldsymbol{\xi}_{i+1}+{\Diamond}_{i+1,n+1}\text{
and }\boldsymbol{\xi}_{i+1}+\sum_{\mathclap{\begin{subarray}{c}k=1\\\ k\neq
i+1,n+1\end{subarray}}}^{n+1}\boldsymbol{\xi}_{k}=\sum_{k=1}^{n}\boldsymbol{\xi}_{k}=-\boldsymbol{\xi}_{n+1},$

the $n$ images of $\boldsymbol{\xi}_{1}+{\Diamond}_{1,n+1}$ under $G_{3}$
share the vertex $-\boldsymbol{\xi}_{n+1}$. ∎

Finally, the next lemma shows that the boundary of
$V\left(\boldsymbol{0}\right)$ is composed of $n(n+1)$ congruent hyper-rhombic
cells associated with the nearest neighbor lattice points of $\mathcal{A}_{n}$
and determines their explicit structures.

######  Lemma 9.

The $(n+1)$ images of $n$ hyper-rhombic cells sharing
$-\boldsymbol{\xi}_{n+1}$ (Lemma 8) under $C_{n+1}$ form the boundary of
$V\left(\boldsymbol{0}\right)$. Overall, the boundary of
$V\left(\boldsymbol{0}\right)$ is composed of $n(n+1)$ congruent hyper-rhombic
faces

$\\{\boldsymbol{\xi}_{i}+{\Diamond}_{ij}:1\leq i\neq j\leq n+1\\}.$

Below are the properties of the face $\boldsymbol{\xi}_{i}+{\Diamond}_{ij}$.
(In the following, we denote $\boldsymbol{\mathit{\Gamma}}_{n,i\bmod
n}=\boldsymbol{\mathit{\Gamma}}_{n,i}$ for simplicity.)

  1. (1)

It is obtained by transforming $\boldsymbol{\xi}_{1}+{\Diamond}_{1,n+1}$ with

$\begin{bmatrix}\boldsymbol{\mathit{\Gamma}}_{n+1,j}\end{bmatrix}\begin{bmatrix}\boldsymbol{\mathit{\Gamma}}_{n,i-j-1}\\\
&1\end{bmatrix}.$

  2. (2)

It is embedded in the bisecting hyperplane of the nearest neighbor point
$\boldsymbol{e}_{i}-\boldsymbol{e}_{j}\in\mathbb{Z}_{\\!\\!\mathcal{A}}^{n}$.

  3. (3)

It is composed of the images of the roof of $\sigma_{0}$ that share the edge
connecting $\boldsymbol{\xi}_{i}$ and $-\boldsymbol{\xi}_{j}$.

###### Proof.

Since (7) (8)

$C_{n+1},G_{2},G_{3}\leq S_{n+1}\text{ and }C_{n+1}\cap G_{2}\cap
G_{3}=\\{\boldsymbol{I}_{n+1}\\},$

the $n(n+1)$ images of $\boldsymbol{\xi}_{1}+{\Diamond}_{1,n+1}$ under
$C_{n+1}G_{3}$ do not overlap. Moreover, since
$|C_{n+1}G_{3}G_{2}|=|S_{n+1}|$, $S_{n+1}$ can be represented as

$S_{n+1}=C_{n+1}G_{3}G_{2}$

and therefore the $n(n+1)$ images of $\boldsymbol{\xi}_{1}+{\Diamond}_{1,n+1}$
under $C_{n+1}G_{3}$ form the boundary of $V\left(\boldsymbol{0}\right)$.

  1. (1)

The claim holds since

$\boldsymbol{\mathit{\Gamma}}_{n+1,j}\begin{bmatrix}\boldsymbol{\mathit{\Gamma}}_{n,i-j-1}\\\
&1\end{bmatrix}(\boldsymbol{\xi}_{1}+{\Diamond}_{1,n+1})=\boldsymbol{\mathit{\Gamma}}_{n+1,j}(\boldsymbol{\xi}_{i-j}+{\Diamond}_{i-j,n+1})=\boldsymbol{\xi}_{i}+{\Diamond}_{i,j}.$

  2. (2)

The claim holds since
$\boldsymbol{\xi}\cdot(\boldsymbol{e}_{i}-\boldsymbol{e}_{j})=0$ for
$\boldsymbol{\xi}\in\boldsymbol{\Xi}_{ij}$.

  3. (3)

Due to Lemma 7, $\boldsymbol{\xi}_{i}+{\Diamond}_{ij}$ is composed of the
$(n-1)!$ simplices sharing the edge connecting

$\boldsymbol{\xi}_{i}\text{ and
}\boldsymbol{\xi}_{i}+\sum_{\mathclap{\begin{subarray}{c}k=1\\\ k\neq
i,j\end{subarray}}}^{n+1}\boldsymbol{\xi}_{k}=-\boldsymbol{\xi}_{j}.$

∎

##  7 Voronoi Cell as the Vertex-First Projection of the Unit Cube

Figure 7:  The orthogonal projection of $\square^{3}$ onto
$\mathbb{H}_{\boldsymbol{1}}$ along $\boldsymbol{1}$ is the Voronoi cell of
$\mathcal{A}_{2}$.

We show that $V\left(\boldsymbol{0}\right)$ is the orthogonal projection of
the $(n+1)$-dimensional unit cube onto $\mathbb{H}_{\boldsymbol{1}}$ along the
‘diagonal direction’ $\boldsymbol{1}$, called the _vertex-first projection_
[5].

######  Lemma 10.

$V\left(\boldsymbol{0}\right)$ is the projection of $\square^{n+1}$ via
$\boldsymbol{\Xi}$. See Figure 7 for $n=2$.

###### Proof.

Note that the $(n+1)$-simplex $\tau_{0}$ (2) is projected to the fundamental
simplex $\sigma_{0}$ (4) since

$\boldsymbol{\Xi}\boldsymbol{u}_{j}=\begin{cases}\boldsymbol{0}=\boldsymbol{v}_{0}&j\in\\{0,n+1\\}\\\
\sum_{i=1}^{j}\boldsymbol{\xi}_{i}=\boldsymbol{v}_{j}&1\leq j\leq
n.\end{cases}$

Specifically, the $n$ vertices $\left\\{\boldsymbol{v}_{j}:1\leq j\leq
n\right\\}$ form the roof of $\sigma_{0}$ (Lemma 3). Since

$\boldsymbol{\Xi}\boldsymbol{\mathit{\Pi}}=\boldsymbol{\mathit{\Pi}}\boldsymbol{\Xi},\quad\boldsymbol{\mathit{\Pi}}\in
S_{n+1}$

and $V\left(\boldsymbol{0}\right)$ is the union of the $(n+1)!$ images of
$\sigma_{0}$ under $S_{n+1}$, the claim holds. ∎

We can deduce that the projections of the vertices of $\square^{n+1}$,
excluding $\boldsymbol{0}$ and $\boldsymbol{1}$, are the vertices of
$V\left(\boldsymbol{0}\right)$ and therefore $V\left(\boldsymbol{0}\right)$ is
the convex hull of these $(2^{n+1}-2)$ vertices.

Alternatively, $V\left(\boldsymbol{0}\right)$ is a _zonotope_ generated by the
line segments $\boldsymbol{\xi}_{1},\dots,\boldsymbol{\xi}_{n+1}.$ [10]:

$V\left(\boldsymbol{0}\right)=\boldsymbol{\Xi}\square^{n+1}=\left\\{\sum_{j=1}^{n+1}t_{j}\boldsymbol{\xi}_{j}:0\leq
t_{j}\leq 1\right\\}.$

Next we show that the $(n+1)$ composing hyper-rhombi of
$V\left(\boldsymbol{0}\right)$ are the projections of the cubic cells of
$\square^{n+1}$ adjacent to the vertex $\boldsymbol{0}$ via $\boldsymbol{\Xi}$
(Figure 7).

######  Lemma 11.

The composing hyper-rhombi of $V\left(\boldsymbol{0}\right)$ (Lemma 5) are the
orthogonal projections of the $n$-dimensional cubic cells of $\square^{n+1}$
adjacent to $\boldsymbol{0}$ via $\boldsymbol{\Xi}$. In other words,

${\Diamond}_{j}=\boldsymbol{\Xi}\square^{n+1}_{j},\quad 1\leq j\leq n+1.$

###### Proof.

The claim holds since

$\displaystyle{\Diamond}_{j}$
$\displaystyle=\left\\{\sum_{\mathclap{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}}^{n+1}t_{i}\boldsymbol{\xi}_{i}:0\leq t_{i}\leq
1\right\\}=\left\\{\sum_{\mathclap{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}}^{n+1}t_{i}\boldsymbol{\Xi}\boldsymbol{e}_{i}:0\leq t_{i}\leq
1\right\\}$
$\displaystyle=\left\\{\boldsymbol{\Xi}\sum_{\mathclap{\begin{subarray}{c}i=1\\\
i\neq j\end{subarray}}}^{n+1}t_{i}\boldsymbol{e}_{i}:0\leq t_{i}\leq
1\right\\}=\boldsymbol{\Xi}\square^{n+1}_{j}.$

∎

Note that the other hyper-rhombic decomposition in (6) are the projection of
the faces of $\square^{n+1}$ via $\boldsymbol{\Xi}$ adjacent to the vertex
$\boldsymbol{1}$ (Figure 7). In other words,

$\boldsymbol{\xi}_{j}+{\Diamond}_{j}=\boldsymbol{\Xi}\left(\boldsymbol{e}_{j}+\square^{n+1}_{j}\right),\quad
1\leq j\leq n+1.$

##  8 Voronoi Cell as the Section of the Voronoi Cell of the
$\mathcal{D}_{n+1}$ Lattice

Figure 8:  The section of the Voronoi cell of the $\mathcal{D}_{3}$ lattice
with $\mathbb{H}_{\boldsymbol{1}}$ is the Voronoi cell of the
$\mathcal{A}_{2}$ lattice.

The following lemma shows that $V\left(\boldsymbol{0}\right)$ is the
intersection of the Voronoi cell of the $\mathcal{D}_{n+1}$ lattice, called
_section_ [5], with $\mathbb{H}_{\boldsymbol{1}}$ in low dimensions.

######  Lemma 12.

Let

$\mathbb{Z}_{\\!\\!\mathcal{D}}^{n+1}:=\left\\{\boldsymbol{x}\in\mathbb{Z}^{n+1}:\boldsymbol{x}\cdot\boldsymbol{1}\text{
is even}\right\\}$

be the $\mathcal{D}_{n+1}$ lattice and
$V_{\mathcal{D}_{n+1}}\left(\boldsymbol{p}\right)$ be the Voronoi cell at
$\boldsymbol{p}\in\mathbb{Z}_{\\!\\!\mathcal{D}}^{n+1}$. In dimensions
$n\in\\{1,2,3\\}$ only, the following holds.

$V\left(\boldsymbol{p}\right)=V_{\mathcal{D}_{n+1}}\left(\boldsymbol{p}\right)\cap\mathbb{H}_{\boldsymbol{1}},\quad\boldsymbol{p}\in\mathbb{Z}_{\\!\\!\mathcal{A}}^{n}.$

###### Proof.

Since

$\mathbb{Z}_{\\!\\!\mathcal{A}}^{n}=\mathbb{Z}^{n+1}\cap\mathbb{H}_{\boldsymbol{1}}=\mathbb{Z}_{\\!\\!\mathcal{D}}^{n+1}\cap\mathbb{H}_{\boldsymbol{1}},$

we only need to show that

$\dim\left(\mathbb{H}_{\boldsymbol{1}}\cap
V_{\mathcal{D}_{n+1}}\left(\boldsymbol{p}\right)\right)<n,\quad\forall\boldsymbol{p}\notin\mathbb{Z}_{\\!\\!\mathcal{A}}^{n}.$

Note that the closest lattice points of $\mathbb{Z}_{\\!\\!\mathcal{D}}^{n+1}$
to $\mathbb{H}_{\boldsymbol{1}}$ which are not on
$\mathbb{H}_{\boldsymbol{1}}$ are those of the form

$\boldsymbol{p}\pm\left(\boldsymbol{e}_{i}+\boldsymbol{e}_{j}\right),\quad\boldsymbol{p}\in\mathbb{Z}_{\\!\\!\mathcal{A}}^{n}\text{
and }1\leq i\neq j\leq n+1.$

Due to the symmetry, we only need to analyze the intersection of
$V_{\mathcal{D}_{n+1}}\left(\boldsymbol{e}_{1}+\boldsymbol{e}_{2}\right)$ with
$\mathbb{H}_{\boldsymbol{1}}$.

It is known that $V_{\mathcal{D}_{n+1}}\left(\boldsymbol{p}\right)$ is the
_pyramidal cube_ [2] whose vertices are

$\left\\{\boldsymbol{p}+\left(\pm\frac{1}{2},\cdots,\pm\frac{1}{2}\right)\right\\}\cup\left\\{\boldsymbol{p}\pm\boldsymbol{e}_{j}:1\leq
j\leq n+1\right\\}$

and therefore for $\boldsymbol{p}=\boldsymbol{e}_{1}+\boldsymbol{e}_{2}$,

$-\frac{n-3}{2}\leq\left((\boldsymbol{e}_{1}+\boldsymbol{e}_{2})+\left(\pm\frac{1}{2},\cdots,\pm\frac{1}{2}\right)\right)\cdot\boldsymbol{1}\leq\frac{n+5}{2}$

and

$\left((\boldsymbol{e}_{1}+\boldsymbol{e}_{2})\pm\boldsymbol{e}_{j}\right)\cdot\boldsymbol{1}\geq
1,\quad 1\leq j\leq n+1.$

Therefore, for $n\in\\{1,2\\}$, all the vertices of
$V_{\mathcal{D}_{n+1}}\left(\boldsymbol{e}_{1}+\boldsymbol{e}_{2}\right)$
belong to the half space
$\\{\boldsymbol{x}\cdot\boldsymbol{1}>0:\boldsymbol{x}\in\mathbb{R}^{n+1}\\}$.
We only need to show that $V_{\mathcal{D}_{4}}\left((1,1,0,0)\right)$
trivially intersect $\mathbb{H}_{\boldsymbol{1}}$, which can be easily
verified since the only vertex of $V_{\mathcal{D}_{4}}\left((1,1,0,0)\right)$
that intersects $\mathbb{H}_{\boldsymbol{1}}$ is
$\left(\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right)$. For $n>3$,
the vertex
$(\frac{1}{2},\frac{1}{2},-\frac{1}{2},\cdots,-\frac{1}{2},)\cdot\boldsymbol{1}<0$
and therefore
$V_{\mathcal{D}_{n+1}}\left(\boldsymbol{e}_{1}+\boldsymbol{e}_{2}\right)$
intersect $\mathbb{H}_{\boldsymbol{1}}$ with $n$-dimensional volume.

∎

Specifically, the Voronoi cell of $\mathbb{Z}_{\\!\\!\mathcal{D}}^{3}$ is the
rhombic dodecahedron whose section with $\mathbb{H}_{\boldsymbol{1}}$ is the
regular hexagon (Figure 8)

$\mathcal{CH}\left(\left\\{\pm\frac{2}{3}\pi\left(2,-1,-1\right)\right\\}\right)$

where $\pi$ denotes all the permutations of the coordinate values. The Voronoi
cell of $\mathbb{Z}_{\\!\\!\mathcal{D}}^{4}$ is the _$24$ -cell_ whose section
with $\mathbb{H}_{\boldsymbol{1}}$ is the rhombic dodecahedron

$\mathcal{CH}\left(\left\\{\frac{1}{2}\pi\left(1,1,-1,-1\right)\right\\}\cup\left\\{\pm\frac{1}{4}\pi(3,-1,-1,-1)\right\\}\right),$

which is transformed to

$\mathcal{CH}\left(\left\\{\pi(\pm
1,0,0)\right\\}\cup\left\\{\left(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2}\right)\right\\}\right)$

by $\boldsymbol{P}_{3}$.

##  9 Conclusion

We characterized the combinatorial structure of the Voronoi cell of the
$\mathcal{A}_{n}$ lattice. Based on the well-known fact that it is composed of
$(n+1)!$ congruent simplices, we showed that it is composed of $(n+1)$
congruent hyper-rhombi. We analyzed the explicit structure of the hyper-
rhombic cells of the Voronoi cell, including the fact that all the
$k$-dimensional faces, $2\leq k\leq n-2$, are hyper-rhombic. We showed that it
is the vertex-first projection of the $(n+1)$-dimensional unit cube and
finally verified that in low dimensions, $n\leq 3$, it is the section of the
Voronoi cell of the $\mathcal{D}_{n+1}$ lattice with the diagonal hyperplane.
All the analyses are done algebraically without any Coxeter-Dynkin diagrams so
they are more accessible to those who are not familiar with them.

## Acknowledgments

This work was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIT) (No. 2021R1F1A1060215).

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