# A Proof of the Weak Simplex Conjecture
††thanks: This article is under peer review.

Adriano Pastore  Information and Signal Processing for Intelligent
Communications
Centre Tecnològic de Telecomunicacions de Catalunya (CTTC)
Av. Carl Friedrich Gauss 7, 08860 Castelldefels, Spain
E-Mail<EMAIL_ADDRESS>
###### Abstract

We solve a long-standing open problem about the optimal codebook structure of
codes in $n$-dimensional Euclidean space that consist of $n+1$ codewords
subject to a per-codeword energy constraint, in terms of minimizing the
average decoding error probability. The conjecture states that optimal
codebooks consist of the $n+1$ vertices of a regular simplex (the
$n$-dimensional generalization of a regular tetrahedron) inscribed in the unit
sphere. A self-contained proof of this conjecture is provided that hinges on
symmetry arguments and leverages a relaxation approach that consists in
_jointly_ optimizing the codebook and the decision regions, rather than the
codeword locations alone.

###### Index Terms:

Sphere packing, Gaussian channel, optimal codes.

##  I Introduction

The fundamental relationships between codebook geometry and decoding error
probability in the classical Gaussian channel are at the core of some
foundational information-theoric questions ever since the early days when
Shannon constructed analytic and geometric approaches to address the problem
of designing codes for communication in noise [1] and to establish links
between communication limits and sphere packing bounds under different types
of power constraints [2]. The case of $n+1$ codewords in $n$-dimensional
Euclidean space is of special theoretic interest in that it corresponds to a
configuration in which the pairwise codeword distance can be set constant (for
any pair of codewords) when choosing the highly symmetric, regular simplex
shape.

The origins of the simplex conjecture can be traced back to Shannon himself,
who suspected that the maximization of minimum codeword distance, though
sensible it may be as an approach to code design, might actually be
suboptimal. Soon the conjecture emerged that, under a per-codeword energy
constraint, the regular simplex inscribed in a centered $(n-1)$-sphere should
indeed be optimal. Balakrishnan produced one of the first comprehensive
studies of this conjecture [3], but could only prove partial results.
Dunbridge [4] showed that the regular simplex is asymptotically optimal at low
noise, and _locally_ optimal at every noise level. Landau and Slepian reported
a full proof [5] as early as 1966, based on geometric arguments involving the
volume of the union of spherical caps, which later turned out to only hold
true for the case $n=3$. The issue that prevented the Landau–Slepian approach
from generalizing to arbitrary $n$ is discussed, e.g., in [6].

The conjecture was further popularized by Tom Cover in 1987 in a famous
compendium of open problems [7] as well as by James Massey’s 1988 Shannon
Lecture [8]. Massey suggested a distinction between two variants of the
conjecture: the classic, so-called Weak Simplex Conjecture (WSC) in which
codewords are subject to a bounded energy constraint, as well as a stronger
claim, which relaxes this constraint to a bound on the sum-energy of all
codewords. The latter variant of the conjecture gained the spotlight and
became known as the Strong Simplex Conjecture (SSC). If proven true, the SSC
would have settled the WSC too. However, the SSC was disproved by Steiner in
1994, who produced a counterexample in form of a signal set construction that
is _not_ inscribed in the sphere and that outperforms the regular simplex [9].

This disproof settled the SSC in the negative, but left the WSC open. Later
analyses such as [10] accrued more evidence in favor of the WSC using
numerical methods. Balitskiy _et al._ [11] gave a modern survey and provided
some equivalent reformulations. Another recent paper, which discusses links to
the closely related Simplex Mean Width conjecture, is [12].

Our proof starts with the original problem formulation and leverages a
relaxation approach which, to the best of our knowledge, was not attempted by
other authors. Before delving into a formal proof of the general case, we will
sketch the general intuition behind our proof method for the three-dimensional
case ($n=3$).

##  II Proof sketch

Consider a problem of optimally placing four points
$\mathcal{W}=(\bm{w}_{1},\bm{w}_{2},\bm{w}_{3},\bm{w}_{4})$ on the unit sphere
$\mathbb{S}_{2}\subset\mathbb{R}^{3}$.111For consistency we have chosen to
denote collections of points as _ordered_ tuples throughout, although in many
cases, the order does not matter. Instances where there is permutation
invariance should be clear from context. It is well known that the regular
tetrahedron inscribed in the sphere is the configuration that maximizes the
minimum distance between any two points. If $\bm{w}$ denotes one of the four
points chosen uniformly at random and $\bm{z}\in\mathbb{R}^{3}$ stands for a
normal Gaussian variable, then the WSC (for $n=3$) consists in the claim that
the regular tetrahedron also maximizes the average probability of correctly
guessing $\bm{w}$ from a noisy observation $\bm{y}=\bm{w}+\sigma\bm{z}$ (where
$\sigma^{2}$ is the noise variance). That is, the regular tetrahedron
maximizes

$Q(\mathcal{W})=\frac{1}{4}\sum_{i=1}^{4}\mathsf{P}\Bigl{\\{}\lVert\bm{y}-\bm{w}_{i}\rVert\leq\min_{j}\lVert\bm{y}-\bm{w}_{j}\rVert\Big{|}\bm{w}=\bm{w}_{i}\Bigr{\\}}.$

This decoding metric naturally partitions $\mathbb{R}^{3}$ into four Voronoi
regions $\mathcal{D}_{1},\dotsc,\mathcal{D}_{4}$ that are flat-faceted cones
with triangular section, whose apices are all located at the origin. These
regions are delimited by bisecting planes that pass through the origin and
whose intersections with $\mathbb{S}_{2}$ form a tiling of the sphere with
four spherical triangles, the vertices of which we denote as
$\mathcal{V}=(\bm{v}_{1},\bm{v}_{2},\bm{v}_{3},\bm{v}_{4})$. The codebook
$\mathcal{W}$ thus fully determines the Voronoi cells $\mathcal{D}_{i}$
(equivalently, the triangle vertices $\mathcal{V}$) and vice-versa. Figure 1
provides an illustration.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$ Figure 1: A codebook
$\mathcal{W}=(\bm{w}_{1},\bm{w}_{2},\bm{w}_{3},\bm{w}_{4})$ for $n=3$, located
on the unit 2-sphere with decision regions delimited by points from
$\mathcal{V}=(\bm{v}_{1},\bm{v}_{2},\bm{v}_{3},\bm{v}_{4})$ forming spherical
triangles.

Now consider a relaxation of the optimization problem where $\mathcal{V}$ and
$\mathcal{W}$ are _not_ mutually determined: instead, we merely impose that
every $\mathcal{D}_{i}$ (more specifically, its associated spherical triangle)
contains a _single_ codeword $\bm{w}_{i}$. In such configurations, different
codewords are _not_ necessarily mirror images of each other along triangle
edges, because we no longer impose that the cones $\mathcal{D}_{i}$ be
delimited by _bisecting_ planes, but instead by any set of planes passing
through the origin that partition $\mathbb{R}^{3}$ into $4$ cones, each
containing a single codeword.

$\bm{w}_{1}$$\bm{w}_{2}$$\bm{w}_{3}$$\bm{v}_{1}$$\bm{v}_{2}$$\bm{v}_{3}$$\mathcal{D}_{2}$$\mathcal{D}_{1}$$\mathcal{D}_{3}$
(a)

$\bm{w}_{1}$$\bm{w}_{2}$$\bm{w}_{3}$$\bm{v}_{1}$$\bm{v}_{2}$$\bm{v}_{3}$$\mathcal{D}_{1}$$\mathcal{D}_{2}$$\mathcal{D}_{3}$
(b)

Figure 2: Fig. 2a shows a sketch (for the case $n=2$ for an easier
visualization) of a codebook $\mathcal{W}$ with its optimal decision regions
delimited by vectors in $\mathcal{V}^{\ast}(\mathcal{W})$. By contrast, Fig.
2b illustrates a valid pair $(\mathcal{W},\mathcal{V})$ under the relaxed
assumptions.

That is, we relax the original optimization problem of the tetrahedron
$\mathcal{W}$, i.e.,

$\max_{\mathcal{W}}Q(\mathcal{W})=\max_{\mathcal{W}}\frac{1}{4}\sum_{i=1}^{4}\mathsf{P}\Bigl{\\{}\bm{y}\in\mathcal{D}_{i}\Big{|}\bm{w}=\bm{w}_{i}\Bigr{\\}}$

to one where we also co-optimize the tetrahedron $\mathcal{V}$, i.e.,

$\max_{\mathcal{W},\mathcal{V}}Q(\mathcal{W},\mathcal{V})=\max_{\mathcal{W},\mathcal{V}}\frac{1}{4}\sum_{i=1}^{4}\mathsf{P}\Bigl{\\{}\bm{y}\in\mathcal{D}(\mathcal{V}_{\setminus
i})\Big{|}\bm{w}=\bm{w}_{i}\Bigr{\\}}.$

Here, $\mathcal{V}_{\setminus i}$ denotes the tuple $\mathcal{V}$ with the
element $\bm{v}_{i}$ removed, and $\mathcal{D}(\mathcal{V}_{\setminus i})$
denotes the cone generated by $\mathcal{V}_{\setminus i}$.

The key reasoning behind proving that regular (equilateral) tetrahedra
$\mathcal{W},\mathcal{V}$ solve this double maximization is that, given a
spherical triangle, e.g., formed by vertices
$(\bm{v}_{2},\bm{v}_{3},\bm{v}_{4})$, the placement of the associated codeword
$\bm{w}_{1}$ on its interior is optimal at the point where
$\mathcal{D}(\bm{v}_{2},\bm{v}_{3},\bm{v}_{4})$ contains the maximum
probability mass of a $\sigma^{2}$-variance Gaussian variable centered at
$\bm{w}_{1}$. This optimal location, which we call the _Gaussian centroid_ of
the spherical triangle formed by vertices
$(\bm{v}_{2},\bm{v}_{3},\bm{v}_{4})$, can be shown to be unique. In other
words, the vertex tuple $(\bm{v}_{2},\bm{v}_{3},\bm{v}_{4})$ uniquely
determines (up to permutation) the optimal codeword $\bm{w}_{1}$. Likewise,
using a proof by contradiction, we can show that fixing two vertices, e.g.,
$(\bm{v}_{2},\bm{v}_{3})$, as well as the centroid $\bm{w}_{1}$, uniquely
determines the third vertex $\bm{v}_{4}$.

Now consider any two adjacent spherical triangles, e.g.,
$(\bm{v}_{2},\bm{v}_{3},\bm{v}_{4})$ and $(\bm{v}_{1},\bm{v}_{2},\bm{v}_{3})$.
We know that in an _optimal_ configuration
$(\mathcal{W}^{\ast},\mathcal{V}^{\ast})$, their respective codewords
$\bm{w}_{1}^{\ast}$ and $\bm{w}_{4}^{\ast}$ must be mirrors of each other
along the common edge (i.e., along the plane spanned by
$(\bm{v}_{2}^{\ast},\bm{v}_{3}^{\ast})$ and the origin), and we also know
that, as argued before, this common edge and the mirrored centroids
$\bm{w}_{1}^{\ast}$ and $\bm{w}_{4}^{\ast}$ determine the vertices
$\bm{v}_{1}^{\ast}$ and $\bm{v}_{4}^{\ast}$. This means that the spherical
triangles $(\bm{v}_{2}^{\ast},\bm{v}_{3}^{\ast},\bm{v}_{4}^{\ast})$ and
$(\bm{v}_{1}^{\ast},\bm{v}_{2}^{\ast},\bm{v}_{3}^{\ast})$ are reflections of
each other along the common edge $(\bm{v}_{2}^{\ast},\bm{v}_{3}^{\ast})$.
Since this reasoning applies to all pairs of triangles, we conclude that all
four triangles that tile $\mathbb{S}_{2}$ are congruent and mutual reflections
of each other, leading to $\mathcal{V}^{\ast}$ (and consequently
$\mathcal{W}^{\ast}$) being a regular tetrahedron.

Yet another way of looking at the problem is that the relaxed double
maximization could be addressed by an _alternating_ optimization of the
centroids $\mathcal{W}$ (for fixed $\mathcal{V}$) and of the spherical
triangles $\mathcal{V}$ (for fixed $\mathcal{W}$) in a way that is reminiscent
of the Lloyd-Max algorithm. We prove by an argument of symmetry that regular
tetrahedra are the only possible point of equilibrium.

##  III Formal proof

Consider a spherical codebook $\mathcal{W}=(\bm{w}_{1},\dotsc,\bm{w}_{n+1})$
of $n+1$ distinct unit-norm codewords from the $n$-dimensional Euclidean space
$\mathbb{R}^{n}$ equipped with the standard scalar product
$\langle\cdot,\cdot\rangle$. The codewords lie on the $(n-1)$-dimensional unit
sphere222We will omit the subscript index used in the usual notation
$\mathcal{S}_{n-1}$, since it will be $n-1$ in all cases.

$\mathbb{S}=\bigl{\\{}\bm{w}\in\mathbb{R}^{n}\colon\lVert\bm{w}\rVert=1\bigr{\\}}$
(1)

where $\lVert\cdot\rVert$ denotes the standard Euclidean norm. Let us further
define the $n$-dimensional Gaussian measure $\gamma$ with density

$g(\bm{x})=\frac{1}{(2\pi)^{n/2}}e^{-\frac{1}{2}\lVert\bm{x}\rVert^{2}}.$ (2)

Next we define a set of $n+1$ disjoint _open_ decision regions
$\mathcal{D}_{1},\dotsc,\mathcal{D}_{n+1}$ (where $\mathcal{D}_{i}$ contains
codeword $\bm{w}_{i}$) whose closures
$\overline{\mathcal{D}}_{i}=\mathcal{D}_{i}\cup\partial\mathcal{D}_{i}$ cover
the Euclidean space
$\mathbb{R}^{n}=\bigcup_{i=1}^{n+1}\overline{\mathcal{D}}_{i}$. Furthermore,
we establish the decoding rule in such way that the decoder declares codeword
$\bm{w}_{i}$ whenever the noisy vector $\bm{y}=\bm{w}+\sigma\bm{z}$ falls
within the decision region $\mathcal{D}_{i}$. The average probability
$Q(\mathcal{W})$ of correct decoding is then

$Q(\mathcal{W})=\frac{1}{n+1}\sum_{i=1}^{n+1}\mathsf{P}\bigl{\\{}\bm{y}\in\mathcal{D}_{i}\big{|}\bm{w}=\bm{w}_{i}\bigr{\\}}$
(3)

where each of the summands can be expressed as an integral

$\displaystyle\mathsf{P}\bigl{\\{}\bm{y}\in\mathcal{D}_{i}\big{|}\bm{w}=\bm{w}_{i}\bigr{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sigma^{n}}\int_{\mathcal{D}_{i}}g{\left(\frac{\bm{x}-\bm{w}_{i}}{\sigma}\right)}\mathrm{d}\bm{x}.$
(4)

###  III-A Geometry of optimal decision regions

As a consequence of the Gaussian density (2) being a monotone function of the
norm $\lVert\bm{x}\rVert$, we have that, given any codebook $\mathcal{W}$, the
optimal (open) decision regions $\mathcal{D}_{i}$, in terms of maximizing
$Q(\mathcal{W})$, are the sets of points closer to $\bm{w}_{i}$ (in Euclidean
distance) than to any other codeword. This well-known fact translates into
$\mathcal{D}_{i}$ being open Voronoi cells pertaining respectively to
codewords $\bm{w}_{i}$. Since the $\bm{w}_{i}$ lie on the unit sphere, each
$\mathcal{D}_{i}$ is the intersection of $n$ open half spaces
$\\{\bm{x}\in\mathbb{R}^{n}\colon\langle\bm{w}_{i}-\bm{w}_{j},\bm{x}\rangle>0\\}$
delimited by planes cutting through the origin. That is, the decision regions
$\mathcal{D}_{i}$ can be represented as

$\displaystyle\mathcal{D}_{i}$ $\displaystyle=$
$\displaystyle\left\\{\bm{x}\in\mathbb{R}^{n}\colon\bm{W}_{i}\bm{x}>\bm{0}\right\\},\quad
i\in[1:n{+}1]$ (5)

where the rows of the square matrix $\bm{W}_{i}\in\mathbb{R}^{n\times n}$ are
given by codeword differences $(\bm{w}_{i}-\bm{w}_{j})^{\mathsf{T}},\
j\in[1:n{+}1]\setminus\\{i\\}$ and the vector inequality in (5) shall mean
that inequality holds for all vector entries simultaneously. Here, $[1:n{+}1]$
denotes the integer range $[1:n{+}1]$.

In the following, we will first assume that codewords are _affinely
independent_ , i.e., that any matrix $\bm{W}_{i},i=[1:n{+}1]$ is full-rank.
Under this assumption, the parameter $\bm{x}\in\mathbb{R}^{n}$ in (5) may be
substituted by $\bm{\bar{\alpha}}\in\mathbb{R}^{n}$ via
$\bm{x}=\bm{W}_{i}^{-1}\bm{\bar{\alpha}}$, which leads to a new
parametrization of decision regions:

$\displaystyle\mathcal{D}_{i}$ $\displaystyle=$
$\displaystyle\left\\{\bm{W}_{i}^{-1}\bm{\bar{\alpha}}\colon\bm{\bar{\alpha}}>\bm{0}\right\\},\quad
i\in[1:n{+}1].$

If we normalize the columns of $\bm{W}_{i}^{-1}$ to unit length and denote the
resulting column vectors as $\bm{v}_{j}^{(i)}\in\mathbb{S},\
j\in[1:n{+}1]\setminus\\{i\\}$, then $\mathcal{D}_{i}$ can be simply
represented as the cone generated by this collection of $n$ unit-length
vectors. Furthermore, by rearranging equations it is easy to check that
$\bm{W}_{i}^{-1}$ and $\bm{W}_{i^{\prime}}^{-1}$ (with $i\neq i^{\prime}$)
have $n-1$ columns in common, which after normalization turn out to be
pairwise equal, in such manner that
$\bm{v}_{j}^{(i)}=\bm{v}_{j}^{(i^{\prime})}$. Therefore, the superscript
‘$(i)$’ can be dropped from notation, so we may define a tuple
$\mathcal{V}=(\bm{v}_{1},\dotsc,\bm{v}_{n+1})$ of unit vectors that constitute
corner points delimiting the conical decision regions
$\mathcal{D}_{i}=\mathcal{D}(\mathcal{V}_{\setminus i})$, namely,

$\displaystyle\mathcal{D}(\mathcal{V}_{\setminus i})$ $\displaystyle=$
$\displaystyle\left\\{{\sum}_{j\neq
i}\alpha_{j}\bm{v}_{j}\colon\bm{\alpha}>\bm{0}\right\\},\quad i\in[1:n{+}1]$

where $\bm{\alpha}\in\mathbb{R}_{+}^{n}$ is the vector of coordinates
$\alpha_{j},\ j\in[1:n{+}1]\setminus\\{i\\}$ and $\mathcal{V}_{\setminus i}$
is shorthand for $\mathcal{V}$ with the $i$-th entry (i.e., $\bm{v}_{i}$)
removed.

In conclusion, we can say that, given any tuple of $n+1$ affinely independent
codewords $\mathcal{W}$, the _optimal_ tuple of $n$ so-called _vertices_
$\mathcal{V}$ delimiting the optimal decision regions $\mathcal{D}_{i}$, which
we may as well denote as $\mathcal{V}^{\ast}(\mathcal{W})$, is fully and
uniquely determined by $\mathcal{W}$ via the above described construction.
Note that by this same construction, it is straightforward to conclude that
the vectors in $\mathcal{V}^{\ast}(\mathcal{W})$ are affinely independent too.
By contrast, the case of _affinely dependent_ codebooks can be discarded
without loss of generality, since affinely dependent codewords would entail
linearly dependent $\mathcal{V}_{\setminus i}$, which corresponds to a
suboptimal configuration according to Farber’s argument laid out in [6, p. 11
and following].

###  III-B The relaxation approach

Let us now consider a relaxation where $\mathcal{V}$ is _not_ determined by
the codebook $\mathcal{W}$ in the way described in the previous section, i.e.,
we do not necessarily set $\mathcal{V}$ to be equal to
$\mathcal{V}^{\ast}(\mathcal{W})$, hence the $\mathcal{D}_{i}$ (which is
shorthand for $\mathcal{D}(\mathcal{V}_{\setminus i})$) are not necessarily
chosen to be optimal decision regions. Instead, any pair
$(\mathcal{W},\mathcal{V})$ may be chosen _freely_ , on the proviso that it
retains the following properties, which are characteristic of any optimality
candidate $(\mathcal{W},\mathcal{V}^{\ast}(\mathcal{W}))$:

  1. 1.

each spherical $(n-1)$-simplex formed by $n$ vertices $\mathcal{V}_{\setminus
i}$ contains exactly one codeword $\bm{w}_{i}$ in its interior;

  2. 2.

$\mathcal{W}$ and $\mathcal{V}$ are sets of affinely independent vectors.

With any valid pair $(\mathcal{W},\mathcal{V})$ satisfying these two
constraints, we can rewrite the maximization objective as

$Q(\mathcal{W},\mathcal{V})=\frac{1}{n+1}\sum_{i=1}^{n+1}\mathsf{P}\bigl{\\{}\bm{y}\in\mathcal{D}(\mathcal{V}_{\setminus
i})\big{|}\bm{w}=\bm{w}_{i}\bigr{\\}}.$ (6)

The relaxation of the original problem $\max_{\mathcal{W}}Q(\mathcal{W})$ now
consists in solving
$\max_{(\mathcal{W},\mathcal{V})}Q(\mathcal{W},\mathcal{V})$ over all pairs of
compatible $\mathcal{W}$ and $\mathcal{V}$. So far, we have elicited that any
jointly optimal pair $(\mathcal{W}^{\ast},\mathcal{V}^{\ast})$ of the relaxed
problem must satisfy the marginal optimality
$\mathcal{V}^{\ast}=\mathcal{V}^{\ast}(\mathcal{W}^{\ast})$ where the optimal
vertex construction $\mathcal{V}^{\ast}(\mathcal{W})$ is as discussed earlier.
Next, to further characterize some key properties of any maximizing pair
$(\mathcal{W}^{\ast},\mathcal{V}^{\ast})$, we move on to discuss properties of
the marginally optimal codebook construction
$\mathcal{W}^{\ast}(\mathcal{V})$. For this purpose, we introduce the concept
of _Gaussian centroid_.

###### Definition 1

A (spherical) Gaussian centroid $\bm{c}\in\mathbb{S}$ of a measurable
spherical set $\mathcal{S}\subset\mathbb{S}$ with parameter $\sigma>0$ is a
unit-norm vector at which a Gaussian variable
$\bm{g}_{\bm{c}}\sim\mathcal{N}(\bm{c},\sigma^{2}\mathbf{I}_{n})$ is centered
so as to maximize the probability
$\mathsf{P}\\{\bm{g}_{\bm{c}}\in\mathcal{C}(\mathcal{S})\\}$, where
$\mathcal{C}(\mathcal{S})=\\{\alpha\bm{s}\colon\alpha\geq
0,\bm{s}\in\mathcal{S}\\}$ denotes the cone generated by $\mathcal{S}$.

We are now ready to articulate two _necessary_ conditions that any jointly
optimal pair $(\mathcal{W}^{\ast},\mathcal{V}^{\ast})$ must necessarily
satisfy:

  1. 1.

For any fixed $\mathcal{W}$, the marginally optimal
$\mathcal{V}^{\ast}(\mathcal{W})$ is such that any hyperplane spanned by a
sub-tuple $\mathcal{V}^{\ast}(\mathcal{W})_{\setminus i,\setminus j}$ bisects
the connecting segment between $\bm{w}_{i}$ and $\bm{w}_{j}$. In other words,
$\bm{w}_{i}$ and $\bm{w}_{j}$ are mirror images of each other with respect to
said bisecting hyperplane.

  2. 2.

For any fixed $\mathcal{V}$, the marginally optimal
$\mathcal{W}^{\ast}(\mathcal{V})$ is such that for each cone
$\mathcal{D}(\mathcal{V}_{\setminus i})$, the corresponding codeword
$\bm{w}_{i}\in\mathcal{D}(\mathcal{V}_{\setminus i})$ is a _Gaussian centroid_
of the spherical $(n-1)$-simplex $\mathcal{D}(\mathcal{V}_{\setminus
i})\cap\mathbb{S}$.

The first of these conditions is well known and was discussed in detail in the
previous section as an immediate consequence of the optimality of nearest-
neighbor decoding. The second follows straightforwardly from Definition 1,
since the placement of each codeword $\bm{w}_{i}$ at a centroid location
guarantees that the $i$-th probability term in the sum (6) is maximized,
without impact on the other summands.

###  III-C Uniqueness of Gaussian centroids

We will show that any spherical $n$-simplex has a _unique_ Gaussian centroid,
which is thus fully determined by the vertices of that simplex. First, recall
that a Gaussian centroid of $\mathcal{D}(\mathcal{V}_{\setminus
i})\cap\mathbb{S}$ does by definition maximize the integral expression

$\displaystyle Q_{i}(\bm{c})$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma^{n}}\int_{\mathcal{D}_{i}}g{\left(\frac{\bm{x}-\bm{c}}{\sigma}\right)}\mathrm{d}\bm{x}$
(7)

subject to $\lVert\bm{c}\rVert=1$. Hence the gradient

$\displaystyle\nabla Q_{i}(\bm{c})$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma^{n}}\int_{\mathcal{D}_{i}}\nabla_{\\!\bm{c}}\,g{\left(\frac{\bm{x}-\bm{c}}{\sigma}\right)}\mathrm{d}\bm{x}$
$\displaystyle=$
$\displaystyle\frac{1}{\sigma^{n+2}}\int_{\mathcal{D}_{i}}(\bm{x}-\bm{c})g{\left(\frac{\bm{x}-\bm{c}}{\sigma}\right)}\mathrm{d}\bm{x}$
$\displaystyle=$
$\displaystyle\frac{\mathsf{P}\\{\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\\}}{\sigma^{2}}\Bigl{(}\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}-\bm{c}\Bigr{)}$

where $\bm{g}_{\bm{c}}$ stands for a $\bm{c}$-centered Gaussian
$\bm{g}_{\bm{c}}\sim\mathcal{N}(\bm{c},\sigma^{2}\mathbf{I}_{n})$, must be
collinear with $\bm{c}$ at a centroid location (i.e., the gradient has only
radial, but no tangential components on the sphere at a centroid location),
which in turn implies that
$\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}$
must be collinear with $\bm{c}$. Factoring in the unit-norm constraint
$\lVert\bm{c}\rVert=1$, we conclude that an optimal $\bm{c}$ must satisfy the
fixed-point equation $\bm{c}=\mathbf{f}_{i}(\bm{c})$, where

$\mathbf{f}_{i}\colon\ \mathbb{S}\to\mathbb{S},\
\bm{c}\mapsto\frac{\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}}{\big{\lVert}\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}\big{\rVert}}.$
(8)

To prove that any fixed-point iteration
$\bm{c}_{m+1}=\mathbf{f}_{i}(\bm{c}_{m}),m\in\mathbb{N}$ converges, we proceed
to showing that the self-map $\mathbf{f}_{i}$ is contractive. For this we
represent $\mathbf{f}_{i}=\mathbf{g}\circ\mathbf{h}_{i}$ as the concatenation
of $\mathbf{g}(\bm{x})=\bm{x}/\lVert\bm{x}\rVert$ and
$\mathbf{h}_{i}(\bm{c})=\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}$,
such that by the chain rule, the Jacobian

$\bm{J}\mathbf{f}_{i}(\bm{c})=\bm{J}(\mathbf{g}\circ\mathbf{h}_{i})(\bm{c})=\bm{J}\mathbf{g}(\mathbf{h}_{i}(\bm{c}))\cdot\bm{J}\mathbf{h}_{i}(\bm{c})$
(9)

factorizes as the matrix product of two Jacobians, namely

$\displaystyle\bm{J}\mathbf{g}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{1}{\lVert\bm{x}\rVert}\left(\mathbf{I}-\frac{\bm{x}}{\lVert\bm{x}\rVert}\frac{\bm{x}^{\mathsf{T}}}{\lVert\bm{x}\rVert}\right)$
(10a) $\displaystyle\bm{J}\mathbf{h}_{i}(\bm{c})$ $\displaystyle=$
$\displaystyle-\frac{1}{\sigma^{2}}\operatorname{cov}(\bm{g}_{\bm{c}}|\bm{g}_{\bm{c}}\in\mathcal{D}_{i}).$
(10b)

Expression (10b) can be derived after some algebra. Indeed, if we denote the
measure of $\bm{g}_{\bm{c}}\sim\mathcal{N}(\bm{c},\sigma^{2}\mathbf{I}_{n})$
as $\gamma_{\bm{c}}$ and abbreviate its differential element as
$\mathrm{d}\gamma_{\bm{c}}(\bm{x})=g{\left(\frac{\bm{x}-\bm{c}}{\sigma}\right)}\mathrm{d}\bm{x}$,

$\displaystyle\IEEEeqnarraymulticol{3}{l}{\bm{J}_{\bm{c}}\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sigma^{n}\gamma_{\bm{c}}(\mathcal{D}_{i})}\int_{\mathcal{D}_{i}}\bm{x}\bm{\nabla}_{\\!\bm{c}}^{\mathsf{T}}\mathrm{d}\gamma_{\bm{c}}(\bm{x})$
$\displaystyle{}-\frac{1}{\sigma^{2n}\gamma_{\bm{c}}(\mathcal{D}_{i})^{2}}\int_{\mathcal{D}_{i}}\bm{x}\mathrm{d}\gamma_{\bm{c}}(\bm{x})\int_{\mathcal{D}_{i}}\bm{\nabla}_{\\!\bm{c}}^{\mathsf{T}}\mathrm{d}\gamma_{\bm{c}}(\bm{x})$
$\displaystyle=$
$\displaystyle-\frac{1}{\sigma^{n+2}\gamma_{\bm{c}}(\mathcal{D}_{i})}\int_{\mathcal{D}_{i}}\bm{x}(\bm{x}-\bm{c})^{\mathsf{T}}\mathrm{d}\gamma_{\bm{c}}(\bm{x})$
$\displaystyle{}+\frac{1}{\sigma^{2n+2}\gamma_{\bm{c}}(\mathcal{D}_{i})^{2}}\int_{\mathcal{D}_{i}}\bm{x}\mathrm{d}\gamma_{\bm{c}}(\bm{x})\int_{\mathcal{D}_{i}}(\bm{x}-\bm{c})^{\mathsf{T}}\mathrm{d}\gamma_{\bm{c}}(\bm{x})$
$\displaystyle=$
$\displaystyle-\frac{1}{\sigma^{2}}\Bigl{(}\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\bm{g}_{\bm{c}}^{\mathsf{T}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}-\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}\bm{c}^{\mathsf{T}}\Bigr{)}$
$\displaystyle{}+\frac{1}{\sigma^{2}}\Bigl{(}\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}\bigl{(}\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}^{\mathsf{T}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}-\bm{c}^{\mathsf{T}}\bigr{)}\Bigr{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{\sigma^{2}}\Bigl{(}\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\bm{g}_{\bm{c}}^{\mathsf{T}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}+\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}^{\mathsf{T}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}\Bigr{)}$

whence (10b).

From [13, Cor. 2.1] and the convexity of $\mathcal{D}_{i}$ we infer that
$\operatorname{cov}(\bm{g}_{\bm{c}})-\operatorname{cov}(\bm{g}_{\bm{c}}|\bm{g}_{\bm{c}}\in\mathcal{D}_{i})$
is positive semidefinite. Since
$\operatorname{cov}(\bm{g}_{\bm{c}})=\sigma^{2}\mathbf{I}$, we further
conclude that $\bm{J}\mathbf{h}_{i}(\bm{c})$ has eigenvalues with absolute
value no larger than unity. Likewise, the projector
$\mathbf{I}-\bm{x}\bm{x}^{\mathsf{T}}/\lVert\bm{x}\rVert^{2}$ appearing in
(10a) has eigenvalues confined on the unit interval. To finish proving that
the product of Jacobians $\bm{J}\mathbf{f}_{i}(\bm{c})$ [cf. (9)] is a
(strictly) contractive mapping, we will show that the scalar factor in (10a)
is smaller than one. That is, we will show that
$\big{\lVert}\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}\big{\rVert}>1$.

It will suffice to prove the stronger result
$\mathsf{E}\bigl{[}\bm{c}^{\mathsf{T}}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}>1$
(recall that $\lVert\bm{c}\rVert=1$). Note that $\mathcal{V}_{\setminus i}$ is
a collection of $n$ linearly independent unit vectors, since the $n+1$ vectors
in $\mathcal{V}$ are assumed affinely independent. Notice also that it
suffices to consider $\bm{c}$ belonging to the _interior_ of the cone
$\mathcal{D}_{i}=\mathcal{D}(\mathcal{V}_{\setminus i})$, i.e.,
$\bm{c}=\sum_{j=1}^{n}\beta_{i}\bm{v}_{j}$ with positive coefficients
$\beta_{i}>0$. As a consequence, any subset of size $n-1$ of the projected
vectors $\bm{\Pi}_{\bm{c}}\bm{v}_{j},j=1,\dotsc,n$ (where
$\bm{\Pi}_{\bm{c}}=\mathbf{I}-\bm{c}\bm{c}^{\mathsf{T}}$) are linearly
independent among them. Therefore, the matrix
$\bm{A}^{-1}=\begin{bmatrix}\bm{c}&\bm{\Pi}_{\bm{c}}\bm{v}_{2}&\ldots&\bm{\Pi}_{\bm{c}}\bm{v}_{n}\end{bmatrix}^{\mathsf{T}}\in\mathbb{R}^{n\times
n}$ (or in short
$\bm{A}^{-1}=\begin{bmatrix}\bm{c}&\bm{\Pi}_{\bm{c}}\bm{V}_{1}\end{bmatrix}^{\mathsf{T}}$
with a full column-rank
$\bm{\Pi}_{\bm{c}}\bm{V}_{1}\in\mathbb{R}^{n\times(n-1)}$) is invertible. Let
us define the bijective linear transformations

$\displaystyle\bm{u}(\bm{x})$
$\displaystyle=\bm{A}^{-1}\frac{\bm{x}-\bm{c}}{\sigma}$
$\displaystyle\bm{x}(\bm{u})$ $\displaystyle=\sigma\bm{A}\bm{u}+\bm{c}$ (11)

such that, by a change of integration variable,

$\displaystyle\mathsf{E}\bigl{[}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}$
$\displaystyle=$
$\displaystyle\frac{\int_{\mathbb{R}^{n}}\bm{x}(\bm{u})g{(\bm{A}\bm{u})}\mathds{1}\bigl{\\{}\bm{x}(\bm{u})\in\mathcal{D}_{i}\bigr{\\}}\mathrm{d}\bm{u}}{\int_{\mathbb{R}^{n}}g{(\bm{A}\bm{u})}\mathds{1}\bigl{\\{}\bm{x}(\bm{u})\in\mathcal{D}_{i}\bigr{\\}}\mathrm{d}\bm{u}}$

where $\mathds{1}\\{\cdot\\}$ denotes the indicator function. Using the latter
expression and (11), we see that proving
$\mathsf{E}\bigl{[}\bm{c}^{\mathsf{T}}\bm{g}_{\bm{c}}\big{|}\bm{g}_{\bm{c}}\in\mathcal{D}_{i}\bigr{]}>1$
boils down to proving

$\int_{\mathbb{R}^{n}}\bm{c}^{\mathsf{T}}\bm{A}\bm{u}\,g{(\bm{A}\bm{u})}\,\mathds{1}\bigl{\\{}\bm{x}(\bm{u})\in\mathcal{D}_{i}\bigr{\\}}\mathrm{d}\bm{u}>0.$
(12)

Note that $\bm{c}^{\mathsf{T}}\bm{A}\bm{u}=u_{1}$. This is because, given how
$\bm{A}^{-1}$ was defined, we have
$\bm{A}=\begin{bmatrix}\bm{c}&\bm{\Pi}_{\bm{c}}\bm{V}_{1}(\bm{V}_{1}^{\mathsf{T}}\bm{\Pi}_{\bm{c}}\bm{V}_{1})^{-1}\end{bmatrix}$,
due to which $\bm{c}^{\mathsf{T}}\bm{A}$ is the first canonical basis (row)
vector. Furthermore, if we denote by $\bm{u}^{-}$ the vector $\bm{u}$ with
only the sign of its first entry $u_{1}$ flipped, then note that the Gaussian
density term satisfies $g{(\bm{A}\bm{u})}=g{(\bm{A}\bm{u}^{-})}$, because
$g{(\bm{A}\bm{u})}=(2\pi)^{-n}\exp\bigl{(}-\bm{u}^{\mathsf{T}}\bm{A}^{\mathsf{T}}\bm{A}\bm{u}/2\bigr{)}$
depends on $u_{1}$ only via its square $u_{1}^{2}$, due to

$\bm{A}^{\mathsf{T}}\bm{A}=\begin{bmatrix}1&\bm{0}_{1\times(n-1)}\\\
\bm{0}_{(n-1)\times
1}&(\bm{V}_{1}^{\mathsf{T}}\bm{\Pi}_{\bm{c}}\bm{V}_{1})^{-1}\end{bmatrix}.$
(13)

In (12), upon splitting the integration over $u_{1}$ between positive and
negative half axes $(-\infty,0]$ and $[0,+\infty)$, then flipping the
integration domain of the first integral, $(-\infty,0]$, by substitution into
$[0,+\infty)$, then eventually reassembling both integrals into one, we can
turn the left-hand side of (12) into

$\int_{\mathbb{R}^{n}}u_{1}g{(\bm{A}\bm{u})}\mathds{1}\bigl{\\{}\bm{x}(\bm{u})\in\mathcal{D}_{i}\bigr{\\}}\mathrm{d}\bm{u}\\\
=\int_{0}^{\infty}\int_{-\infty}^{\infty}\dotsi\int_{-\infty}^{\infty}u_{1}g{(\bm{A}\bm{u})}\nu(\bm{u})\,\mathrm{d}u_{n}\mathrm{d}u_{n-1}\dotso\mathrm{d}u_{1}$
(14)

where
$\nu(\bm{u})=\mathds{1}\bigl{\\{}\bm{x}(\bm{u})\in\mathcal{D}_{i}\bigr{\\}}-\mathds{1}\bigl{\\{}\bm{x}(\bm{u}^{-})\in\mathcal{D}_{i}\bigr{\\}}$.
The function $\nu(\bm{u})$ is non-negative because
$\bm{x}(\bm{u})-\bm{x}(\bm{u}^{-})=\sigma\bm{A}(\bm{u}-\bm{u}^{-})=2\sigma\bm{c}u_{1}$
is a non-negative multiple of $\bm{c}$, and therefore
$\bm{x}(\bm{u}^{-})\in\mathcal{D}_{i}$ implies
$\bm{x}(\bm{u})\in\mathcal{D}_{i}$, due to the conical shape of
$\mathcal{D}_{i}$ and the fact that $\bm{c}\in\mathcal{D}_{i}$. It immediately
follows that (12) holds, yet with a _weak_ inequality.

To strengthen this to a _strict_ inequality as stated in (12), let us run a
closer analysis of the term $\nu(\bm{u})$. Consider the shifted cone
$\hat{\mathcal{D}}_{i}=\mathcal{D}_{i}-2\sigma\bm{c}u_{1}$. Since for
$u_{1}>0$ the shift vector $2\sigma\bm{c}u_{1}$ lies in the _interior_ of the
original cone $\mathcal{D}_{i}$, it is straightforward to deduce that
$\mathcal{D}_{i}\subset\hat{\mathcal{D}}_{i}$, where the set difference
$\hat{\mathcal{D}}_{i}\setminus\mathcal{D}_{i}$ has positive Gaussian measure.

$\mathcal{D}_{i}$$\bm{c}$$\bm{v}_{1}$$\bm{v}_{2}$$2\sigma\bm{c}u_{1}$$\bar{\mathcal{D}_{i}}$
Figure 3: Cone $\mathcal{D}_{i}$ with its shifted version
$\hat{\mathcal{D}}_{i}$

Reverting the previous substitution by means of (11) on the right-hand side of
(14) yields

$\int_{\mathbb{R}^{n}}u_{1}g{(\bm{A}\bm{u})}\mathds{1}\bigl{\\{}\bm{x}(\bm{u})\in\mathcal{D}_{i}\bigr{\\}}\mathrm{d}\bm{u}=\frac{1}{\sigma|\det(\bm{A})|}\times{}\\\
{}\times\int_{\mathbb{R}^{n}}(\bm{c}^{\mathsf{T}}\bm{x}-1)\mathds{1}\\{\bm{c}^{\mathsf{T}}\bm{x}\geq
1\\}g{\left(\frac{\bm{x}-\bm{c}}{\sigma}\right)}\nu(\bm{u}(\bm{x}))\,\mathrm{d}\bm{x}.$
(15)

Under the assumption that the cone $\mathcal{D}_{i}$ is strictly inscribed in
a half space (which we may assume thanks to Balakrishnan’s argument in [3,
Sec. II]), the subset of $\mathbb{R}^{n}$ on which the product
$\mathds{1}\\{\bm{c}^{\mathsf{T}}\bm{x}\geq 1\\}\nu(\bm{u}(\bm{x}))$ is equal
to one, namely the intersection
$(\hat{\mathcal{D}}_{i}\setminus\mathcal{D}_{i})\cap\\{\bm{x}\in\mathbb{R}^{n}\colon\bm{c}^{\mathsf{T}}\bm{x}\geq
1\\}$, as illustrated by the shaded area in Fig. 3, has non-zero volume. Hence
(15) is positive.

This proves (12) and consequently, that $\mathbf{f}_{i}$ is a (strictly)
contractive mapping on the compact closure of $\mathcal{D}_{i}\cap\mathbb{S}$.
By Banach’s fixed-point theorem, the equation $\bm{c}=\mathbf{f}_{i}(\bm{c})$
has a unique solution, which we further know to be an interior point of
$\mathcal{D}_{i}\cap\mathbb{S}$. Therefore, the so-called _Gaussian centroid_
of $\mathcal{D}_{i}\cap\mathbb{S}$ is unique and the optimal codeword
$\bm{w}_{i}$ for a given $\mathcal{V}_{\setminus i}$ is precisely that
Gaussian centroid.

###  III-D One-to-one relationship between vertices and centroids

As was established in the previous section, the Gaussian centroid of spherical
$n$-simplices is unique, so we are legitimized to define a Gaussian centroid
function

$\bm{\zeta}\colon\underbrace{\mathbb{S}\times\dotso\times\mathbb{S}}_{\text{$n$
factors}}\to\mathbb{S}$ (16)

which maps an $n$-tuple of linearly independent unit-norm vertices
$\tilde{\mathcal{V}}=(\tilde{\bm{v}}_{1},\dotsc,\tilde{\bm{v}}_{n})$ to the
Gaussian centroid $\bm{\zeta}(\tilde{\mathcal{V}})$ of the spherical
$n$-simplex spanned by these vertices. We now ask the following question: when
$n-1$ vertices out of the $n$-tuple $\tilde{\mathcal{V}}$ are fixed, as well
as the location of the Gaussian centroid $\bm{\zeta}$, is the remaining vertex
fully determined? To answer this question, consider the _marginal_ centroid
function obtained from $\bm{\zeta}(\cdot)$ when fixing all but one argument.
Since $\bm{\zeta}$ is symmetric, without loss of generality we may consider
the last $n-1$ arguments $\tilde{\mathcal{V}}_{\setminus
1}=(\tilde{\bm{v}}_{2},\dotsc,\tilde{\bm{v}}_{n})$ to be fixed as parameters
and define the marginal centroid function via

$\bm{\zeta}(\cdot;\tilde{\mathcal{V}}_{\setminus 1})\colon\
\mathbb{S}\to\mathbb{S},\
\tilde{\bm{v}}_{1}\mapsto\bm{\zeta}(\tilde{\mathcal{V}}).$ (17)

We are essentially asking whether this self-map is injective for every
linearly independent $\tilde{\mathcal{V}}_{\setminus 1}$. First, let
$\tilde{\bm{e}}$ denote a unit vector generating the null space of
$\tilde{\mathcal{V}}_{\setminus 1}$. Without loss of generality, we can
restrict the domain and co-domain of
$\bm{\zeta}(\cdot;\tilde{\mathcal{V}}_{\setminus 1})$ to the open half-sphere
$\mathbb{S}\cap\mathcal{H}(\tilde{\bm{e}})$ where
$\mathcal{H}(\tilde{\bm{e}})=\\{\bm{x}\colon\langle\tilde{\bm{e}},\bm{x}\rangle>0\\}$,
because it is straightforward to see that the image and pre-image of
$\bm{\zeta}(\cdot;\tilde{\mathcal{V}}_{\setminus 1})$ will either both lie in
$\mathcal{H}(\tilde{\bm{e}})$ or in $\mathcal{H}(-\tilde{\bm{e}})$. All
subsequent reasoning holds for $\tilde{\bm{e}}$ and $-\tilde{\bm{e}}$ alike.

Hence, given two distinct pre-images $\tilde{\bm{v}}_{1}$ and
$\tilde{\bm{v}}_{1}^{\prime}$
($\tilde{\bm{v}}_{1}\neq\tilde{\bm{v}}_{1}^{\prime}$) of the _restricted_
self-map $\bar{\bm{\zeta}}(\cdot;\tilde{\mathcal{V}}_{\setminus
1})\colon\mathbb{S}\cap\mathcal{H}(\tilde{\bm{e}})\to\mathbb{S}\cap\mathcal{H}(\tilde{\bm{e}})$,
we will argue that their images—which for notational brevity we denote as
$\bm{c}=\bm{\zeta}(\tilde{\bm{v}}_{1};\tilde{\mathcal{V}}_{\setminus 1})$ and
$\bm{c}^{\prime}=\bm{\zeta}(\tilde{\bm{v}}_{1}^{\prime};\tilde{\mathcal{V}}_{\setminus
1})$, respectively—are necessarily different. Let us also denote the cones
$\tilde{\mathcal{D}}=\mathcal{D}(\tilde{\mathcal{V}})$ and
$\tilde{\mathcal{D}}^{\prime}=\mathcal{D}(\tilde{\mathcal{V}}^{\prime})$ as
well as their intersection
$\mathcal{I}=\tilde{\mathcal{D}}\cap\tilde{\mathcal{D}}^{\prime}$. We further
introduce

$\displaystyle\bm{i}$ $\displaystyle=$
$\displaystyle\mathsf{E}[\bm{g}_{\bm{c}}|\bm{g}_{\bm{c}}\in\mathcal{I}]$ (18a)
$\displaystyle\bm{i}^{\prime}$ $\displaystyle=$
$\displaystyle\mathsf{E}[\bm{g}_{\bm{c}^{\prime}}|\bm{g}_{\bm{c}^{\prime}}\in\mathcal{I}]$
(18b) $\displaystyle\bm{t}$ $\displaystyle=$
$\displaystyle\mathsf{E}[\bm{g}_{\bm{c}}|\bm{g}_{\bm{c}}\in\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime}]$
if $\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime}\neq\emptyset$,
else $\bm{t}=\bm{0}$ (18c) $\displaystyle\bm{t}^{\prime}$ $\displaystyle=$
$\displaystyle\mathsf{E}[\bm{g}_{\bm{c}^{\prime}}|\bm{g}_{\bm{c}^{\prime}}\in\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}}]$
if $\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}}\neq\emptyset$,
else $\bm{t}^{\prime}=\bm{0}$. (18d)

As previously, we denote the measure of
$\bm{g}_{\bm{c}}\sim\mathcal{N}(\bm{c},\sigma^{2}\mathbf{I}_{n})$ as
$\gamma_{\bm{c}}$. Based on the fixed-point identity that any centroid must
satisfy [cf. Section III-C], $\bm{c}$ and $\bm{c}^{\prime}$ can be expressed
as linear combinations of vectors $\bm{i}$, $\bm{t}$ and $\bm{t}^{\prime}$:

$\displaystyle\bm{c}$ $\displaystyle=$
$\displaystyle\frac{\mathsf{E}[\bm{g}_{\bm{c}}|\bm{g}_{\bm{c}}\in\tilde{\mathcal{D}}]}{\lVert\mathsf{E}[\bm{g}_{\bm{c}}|\bm{g}_{\bm{c}}\in\tilde{\mathcal{D}}]\rVert}$
= γc(I) i \+ γc(~D∖~D’) t ∥γc(I) i \+ γc(~D∖~D’) t ∥ (19a)
$\displaystyle\bm{c}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\mathsf{E}[\bm{g}_{\bm{c}^{\prime}}|\bm{g}_{\bm{c}^{\prime}}\in\tilde{\mathcal{D}}^{\prime}]}{\lVert\mathsf{E}[\bm{g}_{\bm{c}^{\prime}}|\bm{g}_{\bm{c}^{\prime}}\in\tilde{\mathcal{D}}^{\prime}]\rVert}$
= γc’(I) i’ + γc’(~D’ ∖~D) t’ ∥γc’(I) i’ + γc’(~D’ ∖~D) t’ ∥. (19b)

We now perform a proof by contradiction: let us assume that
$\tilde{\bm{v}}_{1}\neq\tilde{\bm{v}}_{1}^{\prime}$ and
$\bm{c}=\bm{c}^{\prime}$. From the former assumption
($\tilde{\bm{v}}_{1}\neq\tilde{\bm{v}}_{1}^{\prime}$), it follows that
$\tilde{\mathcal{D}}\neq\tilde{\mathcal{D}}^{\prime}$ and that at least one of
the two cone differences
$\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime}$ or
$\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}}$ in non-empty hence
at least one of either
$\gamma_{\bm{c}}(\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime})$ or
$\gamma_{\bm{c}^{\prime}}(\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}})$
is positive. Meanwhile, the assumption $\bm{c}=\bm{c}^{\prime}$ also implies
$\bm{i}=\bm{i}^{\prime}$ [cf. (LABEL:def:i)–(LABEL:def:i_prime)] as well as
$\gamma_{\bm{c}}(\mathcal{I})=\gamma_{\bm{c}^{\prime}}(\mathcal{I})$. As a
result, we can replace $\bm{c}^{\prime}$ by $\bm{c}$ and $\bm{i}^{\prime}$ by
$\bm{i}$ in (19a)–(19b). We distinguish two cases:

  * •

If $\bm{i}$, $\bm{t}$ and $\bm{t^{\prime}}$ are linearly _independent_ , then
the right-hand sides of (19a)–(19b) clearly cannot coincide, since we just saw
that either
$\gamma_{\bm{c}}(\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime})\neq
0$ or
$\gamma_{\bm{c}^{\prime}}(\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}})\neq
0$.

  * •

If $\bm{i}$, $\bm{t}$ and $\bm{t^{\prime}}$ are linearly _dependent_ , we
claim that this can only occur if $\bm{t}=\bm{0}$ (resp.
$\bm{t}^{\prime}=\bm{0}$) and $\bm{i},\bm{t}^{\prime}$ (resp. $\bm{i},\bm{t}$)
are non-collinear, or if $\bm{i}=\alpha\bm{t}+\alpha^{\prime}\bm{t}^{\prime}$
is a linear combination of non-collinear $\bm{t}$ and $\bm{t}^{\prime}$ with
_positive_ coefficients $\alpha$ and $\alpha^{\prime}$ (properly scaled such
that $\lVert\bm{i}\rVert=1$). The proof of these claims is deferred to the
Appendix. In the former situation, i.e., $\bm{t}=\bm{0}$ (or
$\bm{t}^{\prime}=\bm{0}$), clearly the right-hand sides of (20a)–(20b) cannot
be equal due to $\bm{t}$ (or $\bm{t}^{\prime}$) not being collinear with
$\bm{i}$. In the latter situation, let us insert
$\bm{i}=\alpha\bm{t}+\alpha^{\prime}\bm{t}^{\prime}$ into the above
expressions (19a) and (19b) and set $\bm{c}^{\prime}=\bm{c}$, then (19a)–(19b)
turn into

$\displaystyle\bm{c}$ $\displaystyle=$
$\displaystyle\frac{\bigl{(}\gamma_{\bm{c}}(\mathcal{I})\alpha+\gamma_{\bm{c}}(\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime})\bigr{)}\bm{t}+\gamma_{\bm{c}}(\mathcal{I})\alpha^{\prime}\bm{t}^{\prime}}{\lVert\bigl{(}\gamma_{\bm{c}}(\mathcal{I})\alpha+\gamma_{\bm{c}}(\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime})\bigr{)}\bm{t}+\gamma_{\bm{c}}(\mathcal{I})\alpha^{\prime}\bm{t}^{\prime}\rVert}$
(20a) $\displaystyle\bm{c}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{\bm{c}}(\mathcal{I})\alpha\bm{t}+\bigl{(}\gamma_{\bm{c}}(\mathcal{I})\alpha^{\prime}+\gamma_{\bm{c}}(\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}})\bigr{)}\bm{t}^{\prime}}{\lVert\gamma_{\bm{c}}(\mathcal{I})\alpha\bm{t}+\bigl{(}\gamma_{\bm{c}}(\mathcal{I})\alpha^{\prime}+\gamma_{\bm{c}}(\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}})\bigr{)}\bm{t}^{\prime}\rVert}.$
(20b)

However, these two equations cannot hold simultaneously: given that the left-
hand sides of (20a)–(20b) are identical, the coefficients of the linearly
independent vectors $\bm{t}$ and $\bm{t}^{\prime}$ on the respective right-
hand sides of (20a)–(20b) would need to be identical too. Yet the ratio
between the coefficients of $\bm{t}$ and $\bm{t}^{\prime}$ is larger for (20a)
than it is for (20b):

$\frac{\gamma_{\bm{c}}(\mathcal{I})\alpha+\gamma_{\bm{c}}(\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime})}{\gamma_{\bm{c}}(\mathcal{I})\alpha^{\prime}}>\frac{\gamma_{\bm{c}}(\mathcal{I})\alpha}{\gamma_{\bm{c}}(\mathcal{I})\alpha^{\prime}+\gamma_{\bm{c}}(\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}})}.$

This inequality is strict because
$\tilde{\bm{v}}_{n}\neq\tilde{\bm{v}}_{n}^{\prime}$ implies that at least one
of the two terms
$\gamma_{\bm{c}}(\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime})$
and
$\gamma_{\bm{c}}(\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}})$ is
positive. Hence, at least one of the two equalities (20a) and (20b) fails to
hold, so we arrive at a contradiction. We can thus conclude that $\bm{c}$ and
$\bm{c}^{\prime}$ are _not_ collinear and must therefore be distinct. The
self-map $\bm{\zeta}(\cdot;\cdot)$ is thus injective. To express this in more
generality, we can conclude that any $(n{-}1)$-tuple $\mathcal{V}_{\setminus
i,\setminus j}$ of vertices belonging to an $n$-tuple $\mathcal{V}_{\setminus
i}$, in conjunction with the locus $\bm{c}$ of its Gaussian centroid, will
fully determine the $j$-th vertex $\bm{v}_{j}$.

###  III-E Assembling it all: tessellating $\mathbb{S}$ with spherical
simplices

We can now finally rely on two necessary conditions for a pair
$(\mathcal{W}^{\ast},\mathcal{V}^{\ast})$ to be optimal:

  * •

Two codewords $\bm{w}_{i}^{\ast}$ and $\bm{w}_{j}^{\ast}$ are symmetric images
of each other along the $(n{-}1)$-dimensional plane passing through the origin
and spanned by the vectors in $\mathcal{V}_{\setminus i,\setminus j}^{\ast}$.

  * •

Given $n-1$ vertices $\mathcal{V}_{\setminus i,\setminus j}^{\ast}$ (which are
necessarily common vertices of the two tuples $\mathcal{V}_{\setminus
i}^{\ast}$ and $\mathcal{V}_{\setminus j}^{\ast}$) and given that the Gaussian
centroids of $\mathcal{V}_{\setminus i}^{\ast}$ and $\mathcal{V}_{\setminus
j}^{\ast}$, namely $\bm{w}_{i}^{\ast}$ and $\bm{w}_{j}^{\ast}$, are also
given, the remaining vertices $\bm{v}_{i}^{\ast}$ and $\bm{v}_{j}^{\ast}$ are
fully determined.

Combining these two conditions, and owing to the fact that $\bm{\zeta}$ is
invariant to isometry, one ends up with a third condition as a consequence:
namely, that any two spherical $n$-simplices like those generated by
$\mathcal{V}_{\setminus i}^{\ast}$ and $\mathcal{V}_{\setminus j}^{\ast}$ are
mutually symmetric images (hence congruent).

We can now argue that $\mathcal{V}^{\ast}$ must be a regular simplex as
follows: consider any two adjacent edges of the $(n{+}1)$-simplex
corresponding to $\mathcal{V}^{\ast}$, say for example the segments
$\bm{v}_{k}^{\ast}\bm{v}_{\ell}^{\ast}$ and
$\bm{v}_{\ell}^{\ast}\bm{v}_{m}^{\ast}$ (with $k\neq m$). Consider the
$(n{-}1)$-simplex $\mathcal{V}_{\setminus k,\setminus m}^{\ast}$ (which
includes the vertex $\bm{v}_{\ell}^{\ast}$). The two $n$-simplices
$\mathcal{V}_{\setminus k}^{\ast}$ and $\mathcal{V}_{\setminus m}^{\ast}$ must
be symmetric along the plane spanned by the common vertices
$\mathcal{V}_{\setminus k,\setminus m}^{\ast}$. Hence the segments
$\bm{v}_{k}^{\ast}\bm{v}_{\ell}^{\ast}$ and
$\bm{v}_{\ell}^{\ast}\bm{v}_{m}^{\ast}$ are mutually symmetric along that
plane, and therefore of equal lengths
$\lVert\bm{v}_{k}^{\ast}-\bm{v}_{\ell}^{\ast}\rVert=\lVert\bm{v}_{\ell}^{\ast}-\bm{v}_{m}^{\ast}\rVert$.
Since this argument applies to any pair of adjacent edges, this means that all
edges of the $(n{+}1)$-simplex $\mathcal{V}^{\ast}$ have equal length, hence
$\mathcal{V}^{\ast}$ is a regular $n$-simplex. By symmetry and isometry of the
centroid function, $\mathcal{W}^{\ast}$ must be a regular simplex too. This
concludes the proof of the Weak Simplex Conjecture.

##  IV Appendix

In the following, we prove two claims from Section III-D:

  1. 1.

$\bm{t}=\bm{0}$ (resp. $\bm{t}^{\prime}=\bm{0}$) implies non-collinearity of
$\bm{i}$ and $\bm{t}^{\prime}$ (resp. $\bm{i}$ and $\bm{t}$);

  2. 2.

$\bm{i}$, $\bm{t}$, $\bm{t^{\prime}}$ can only be linearly dependent if
$\bm{t}$ and $\bm{t}^{\prime}$ are non-collinear and
$\bm{i}=\alpha\bm{t}+\alpha^{\prime}\bm{t}^{\prime}$ with _positive_ $\alpha$
and $\alpha^{\prime}$.

Recall that $\tilde{\mathcal{V}}$ and $\tilde{\mathcal{V}}^{\prime}$ denote
linearly independent $n$-tuples of vertices that differ only on the first
element ($\tilde{\bm{v}}_{1}\neq\tilde{\bm{v}}_{1}^{\prime}$). The cones
$\tilde{\mathcal{D}}$ and $\tilde{\mathcal{D}}^{\prime}$ can be expressed as
intersections $\tilde{\mathcal{D}}=\bigcap_{i=1}^{n}\mathcal{H}_{i}$ and
$\tilde{\mathcal{D}}^{\prime}=\bigcap_{i=1}^{n}\mathcal{H}_{i}^{\prime}$ of
half spaces defined as

$\displaystyle\mathcal{H}_{i}$ $\displaystyle=$
$\displaystyle\bigl{\\{}\bm{x}\in\mathbb{R}^{n}\colon\langle\bm{e}_{i},\bm{x}\rangle>0\bigr{\\}},\quad
i=1,\dotsc,n$ (21a) $\displaystyle\mathcal{H}_{i}^{\prime}$ $\displaystyle=$
$\displaystyle\bigl{\\{}\bm{x}\in\mathbb{R}^{n}\colon\langle\bm{e}_{i}^{\prime},\bm{x}\rangle>0\bigr{\\}},\quad
i=1,\dotsc,n$ (21b)

where $\bm{e}_{i}$ (resp. $\bm{e}_{i}^{\prime}$) are unit vectors, each of
which is chosen orthogonal to the $(n-1)$-dimensional subspace spanned by
$\tilde{\mathcal{V}}_{\setminus i}$ (resp. $\tilde{\mathcal{V}}_{\setminus
i}^{\prime}$) and the orientation of $\bm{e}_{i}$ (resp.
$\bm{e}_{i}^{\prime}$) is chosen such that
$\langle\bm{e}_{i},\tilde{\bm{v}}_{i}\rangle>0$ (resp.
$\langle\bm{e}_{i}^{\prime},\tilde{\bm{v}}_{i}\rangle>0$). Note that
$\bm{e}_{1}=\bm{e}_{1}^{\prime}$ due to common entries in
$\tilde{\mathcal{V}}$ and $\tilde{\mathcal{V}}^{\prime}$, so
$\mathcal{H}_{1}=\mathcal{H}_{1}^{\prime}$.

Note that every triple $(\bm{e}_{1},\bm{e}_{i},\bm{e}_{i}^{\prime})$ for some
$i\in[2{:}n]$ is linearly dependent. To see this, recall that from their
definition, each of the three vectors $\bm{e}_{1}$, $\bm{e}_{i}$ and
$\bm{e}_{i}^{\prime}$ is orthogonal to each of the $n-2$ linearly independent
vectors in $\tilde{\mathcal{V}}_{\setminus 1,\setminus i}$, whose null space
is two-dimensional, hence the three vectors $\bm{e}_{1}$, $\bm{e}_{i}$ and
$\bm{e}_{i}^{\prime}$ must be coplanar (linearly dependent), satisfying
$\lambda_{1}\bm{e}_{1}+\lambda_{i}\bm{e}_{i}+\lambda_{i}^{\prime}\bm{e}_{i}^{\prime}=\bm{0}$
for some coefficients $\lambda_{1}$, $\lambda_{i}$ and $\lambda_{i}^{\prime}$
not all equal to zero. In fact, it turns out that all three coefficients are
non-zero, because $\bm{e}_{1}$, $\bm{e}_{i}$, $\bm{e}_{i}^{\prime}$ are
pairwise linearly independent due to $\tilde{\mathcal{V}}$ being linearly
independent (hence all null spaces of $\tilde{\mathcal{V}}_{\setminus
1,\setminus i}$ are distinct for $i\in[1{:}n]$). Let us therefore set
$\lambda_{1}=1$ without loss of generality.

Upon applying the scalar product with $\tilde{\bm{v}}_{i}$ on both sides of
the identity
$\bm{e}_{1}+\lambda_{i}\bm{e}_{i}+\lambda_{i}^{\prime}\bm{e}_{i}^{\prime}=\bm{0}$
and recalling that $\langle\bm{e}_{1},\tilde{\bm{v}}_{i}\rangle=0$, we get
$\lambda_{i}\langle\bm{e}_{i},\tilde{\bm{v}}_{i}\rangle+\lambda_{i}^{\prime}\langle\bm{e}_{i}^{\prime},\tilde{\bm{v}}_{i}\rangle=0$.
Since the scalar products $\langle\bm{e}_{i},\tilde{\bm{v}}_{i}\rangle$ and
$\langle\bm{e}_{i}^{\prime},\tilde{\bm{v}}_{i}\rangle$ are both positive, it
follows that $\lambda_{i}$ and $\lambda_{i}^{\prime}$ must be of opposite
signs (i.e., $\lambda_{i}\lambda_{i}^{\prime}<0$), so that for each
$i\in[2{:}n]$ we have one of two situations: either $\lambda_{1}=1$,
$\lambda_{i}>0$ and $\lambda_{i}^{\prime}<0$, else $\lambda_{1}=1$,
$\lambda_{i}<0$ and $\lambda_{i}^{\prime}>0$. This allows us to partition the
index set $[2{:}n]$ into a subset $\mathcal{J}$ and its complement
$\bar{\mathcal{J}}=[2{:}n]\setminus\mathcal{J}$, where

$\displaystyle\mathcal{J}$ $\displaystyle=$
$\displaystyle\bigl{\\{}i\in[2{:}n]\colon\lambda_{i}>0,\lambda_{i}^{\prime}<0\bigr{\\}}.$
(22)

If $i\in\mathcal{J}$, then any $\bm{x}\in\mathbb{R}^{n}$ satisfying
$\langle\bm{e}_{1},\bm{x}\rangle>0$ (i.e.,
$\bm{x}\in\mathcal{H}_{1}=\mathcal{H}_{1}^{\prime}$) and
$\langle\bm{e}_{i},\bm{x}\rangle>0$ (i.e., $\bm{x}\in\mathcal{H}_{i}$) will
also satisfy

$\langle\bm{e}_{i}^{\prime},\bm{x}\rangle=-\frac{1}{\lambda_{i}^{\prime}}\langle\bm{e}_{1},\bm{x}\rangle-\frac{\lambda_{i}}{\lambda_{i}^{\prime}}\langle\bm{e}_{i},\bm{x}\rangle>0,$

that is to say, $\bm{x}\in\mathcal{H}_{i}^{\prime}$. In other words, if
$i\in\mathcal{J}$, then
$\mathcal{H}_{1}\cap\mathcal{H}_{i}\subset\mathcal{H}_{1}\cap\mathcal{H}_{i}^{\prime}$.
Conversely, if $i\in\bar{\mathcal{J}}$, the inclusion holds in reverse, i.e.,
$\mathcal{H}_{1}\cap\mathcal{H}_{i}\supset\mathcal{H}_{1}\cap\mathcal{H}_{i}^{\prime}$.
Let $\mathcal{K}_{i}$ and $\mathcal{K}_{i}^{\prime}$ be shorthand for
$\mathcal{H}_{1}\cap\mathcal{H}_{i}$ and
$\mathcal{H}_{1}\cap\mathcal{H}_{i}^{\prime}$, respectively, so
$\tilde{\mathcal{D}}=\bigcap_{i=2}^{n}\mathcal{K}_{i}$ and
$\tilde{\mathcal{D}}^{\prime}=\bigcap_{i=2}^{n}\mathcal{K}_{i}^{\prime}$.
Furthermore, $\mathcal{K}_{i}\subset\mathcal{K}_{i}^{\prime}$ (for
$i\in\mathcal{J}$) and $\mathcal{K}_{i}\supset\mathcal{K}_{i}^{\prime}$ (for
$i\in\bar{\mathcal{J}}$). It follows from these inclusion properties that

$\displaystyle\tilde{\mathcal{D}}\cap\tilde{\mathcal{D}}^{\prime}$
$\displaystyle=$
$\displaystyle\left(\textstyle\bigcap_{i\in\mathcal{J}}\mathcal{K}_{i}\right)\cap\left(\textstyle\bigcap_{i\in\bar{\mathcal{J}}}\mathcal{K}_{i}^{\prime}\right)$
(23a) $\displaystyle\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime}$
$\displaystyle\subset$
$\displaystyle\left(\textstyle\bigcap_{i\in\mathcal{J}}\mathcal{K}_{i}\right)\setminus\left(\textstyle\bigcap_{i\in\bar{\mathcal{J}}}\mathcal{K}_{i}^{\prime}\right)$
(23b) $\displaystyle\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}}$
$\displaystyle\subset$
$\displaystyle\left(\textstyle\bigcap_{i\in\bar{\mathcal{J}}}\mathcal{K}_{i}^{\prime}\right)\setminus\left(\textstyle\bigcap_{i\in\mathcal{J}}\mathcal{K}_{i}\right).$
(23c)

To see why (23b) holds, consider

$\displaystyle\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime}$
$\displaystyle=$
$\displaystyle\left(\textstyle\bigcap_{i=2}^{n}\mathcal{K}_{i}\right)\setminus\left(\left(\textstyle\bigcap_{i\in\bar{\mathcal{J}}}\mathcal{K}_{i}^{\prime}\right)\cap\left(\textstyle\bigcap_{i\in\mathcal{J}}\mathcal{K}_{i}^{\prime}\right)\right)$
$\displaystyle\subset$
$\displaystyle\left(\textstyle\bigcap_{i\in\mathcal{J}}\mathcal{K}_{i}\right)\setminus\left(\left(\textstyle\bigcap_{i\in\bar{\mathcal{J}}}\mathcal{K}_{i}^{\prime}\right)\cap\left(\textstyle\bigcap_{i\in\mathcal{J}}\mathcal{K}_{i}^{\prime}\right)\right)$
$\displaystyle=$
$\displaystyle\left(\textstyle\bigcap_{i\in\mathcal{J}}\mathcal{K}_{i}\right)\setminus\left(\textstyle\bigcap_{i\in\bar{\mathcal{J}}}\mathcal{K}_{i}^{\prime}\right)$

where the last equality follows from
$\textstyle\bigcap_{i\in\mathcal{J}}\mathcal{K}_{i}\subset\textstyle\bigcap_{i\in\mathcal{J}}\mathcal{K}_{i}^{\prime}$.
Property (23c) is derived similarly. The relations (23a)–(23c) imply that
$\bm{i}\in\tilde{\mathcal{D}}\cap\tilde{\mathcal{D}}^{\prime}$,
$\bm{t}\in\tilde{\mathcal{D}}\setminus\tilde{\mathcal{D}}^{\prime}$ and
$\bm{t}^{\prime}\in\tilde{\mathcal{D}}^{\prime}\setminus\tilde{\mathcal{D}}$
satisfy

$\displaystyle\langle\bm{e}_{i},\bm{i}\rangle$ $\displaystyle>$ $\displaystyle
0\ \text{for all $i\in\mathcal{J}$}$ , ⟨e_i’, i ⟩ ¿ 0 for all
$i\in\bar{\mathcal{J}}$ (24a) $\displaystyle\langle\bm{e}_{i},\bm{t}\rangle$
$\displaystyle>$ $\displaystyle 0\ \text{for all $i\in\mathcal{J}$}$ , ⟨e_i’,
t ⟩ ≤ 0 for some $i\in\bar{\mathcal{J}}$ (24b)
$\displaystyle\langle\bm{e}_{i}^{\prime},\bm{t}^{\prime}\rangle$
$\displaystyle>$ $\displaystyle 0\ \text{for all $i\in\bar{\mathcal{J}}$}$ ,
⟨e_i, t’ ⟩ ≤ 0 for some $i\in\mathcal{J}$. (24c)

provided that $\bm{t}\neq\bm{0}$ and $\bm{t}^{\prime}\neq\bm{0}$ [cf.
(18c)–(18d)]. In the contrary case, if $\bm{t}=\bm{0}$ (resp.
$\bm{t}^{\prime}=\bm{0}$), the conditions (24c) (resp. (24b)) and (24a) render
collinearity of $\bm{i}$ and $\bm{t}^{\prime}$ (resp. $\bm{i}$ and
$\bm{t}^{\prime}$) impossible. This establishes the first claim.

If $\bm{t}$ and $\bm{t}^{\prime}$ are both non-zero, we can immediately
conclude that $\bm{i}$, $\bm{t}$ and $\bm{t}^{\prime}$ are pairwise non-
collinear, for otherwise, (24a)–(24c) could not hold simultaneously. Supposing
that $\bm{i}=\alpha\bm{t}+\alpha^{\prime}\bm{t}^{\prime}$, we can outrule that
$\alpha$ and $\alpha^{\prime}$ are both negative, because $\bm{i}$, $\bm{t}$
and $\bm{t}^{\prime}$ belong to the same half space $\mathcal{H}_{1}$. Also,
$\alpha\alpha^{\prime}=0$ is impossible because when contrasting (24b)–(24c)
against (24a), we see $\bm{i}$ cannot be collinear to either $\bm{t}$ or
$\bm{t}^{\prime}$. We can also outrule $\alpha\alpha^{\prime}<0$ because by
(24b)–(24c) there exist indices $j\in\bar{\mathcal{J}}$ and $k\in\mathcal{J}$
such that $\langle\bm{e}_{j}^{\prime},\bm{t}\rangle\leq 0$,
$\langle\bm{e}_{j}^{\prime},\bm{t}^{\prime}\rangle>0$ and
$\langle\bm{e}_{k},\bm{t}^{\prime}\rangle\leq 0$,
$\langle\bm{e}_{k},\bm{t}\rangle>0$. Hence, there exists an index $i$ (equal
to either $j$ or $k$) such that either
$\langle\bm{e}_{i},\bm{i}\rangle=\alpha\langle\bm{e}_{i},\bm{t}\rangle+\alpha^{\prime}\langle\bm{e}_{i},\bm{t}^{\prime}\rangle$
or
$\langle\bm{e}_{i}^{\prime},\bm{i}\rangle=\alpha\langle\bm{e}_{i}^{\prime},\bm{t}\rangle+\alpha^{\prime}\langle\bm{e}_{i}^{\prime},\bm{t}^{\prime}\rangle$
is negative, whereby (24a) is violated. This establishes the second claim.

## Acknowledgments

The author is indebted to Aaron Goldsmith, Alexey Balitskiy and Roman Karasev,
who have pointed out some issues in an earlier draft. Helpful discussions with
them to improve the manuscript, as well as with Michael Gastpar, are
gratefully acknowledged.

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