# On Barker-Larman Conjecture relative to a convex body with centrally
symmetric sections

E. Morales-Amaya
Facultad de Matemáticas-Acapulco,
Universidad Autónoma de Guerrero, México

###### Abstract

Let $K\subset\mathbb{R}^{n}$ be a convex body, $n\geq 3$. We say that $K$
satisfies the Barker-Larman condition if there exists a ball
$B\subset\text{int}K$ such that for every supporting hyperplane $\Pi$ of $B$,
the section $\Pi\cap K$ is a centrally symmetric set. On the other hand, we
say that $K$ satisfies the Montejano condition if there exists a ball
$B\subset\text{int}K$ such that for every supporting hyperplane $\Pi$ of $B$,
the section $\Pi\cap K$ is a body of constant width. In this work we prove the
following results where, in both cases, $K$ is an $O$-symmetric convex body
and $B\subset\text{int}K$ is a ball such that $O\notin B$: 1) If $K$ is
strictly convex body and satisfies the Barker-Larman condition with respect to
$B$, then $K$ is an ellipsoid; 2) If $K$ satisfies the Montejano condition
relative to $B$, then $K$ is a ball.

##  1 Introduction

Let $K\subset\mathbb{R}^{n}$ be a convex body, i.e., a compact and convex set
with non-empty interior. Let $H$ be a plane of dimension $r$, with $1\leq
r\leq n-1$. If $H\cap\text{int}K\neq\emptyset$, then we say that $H\cap K$ is
an $r$-dimensional section of $K$. The following theorem is due to S.
Olovjanishnikov [7]:

Theorem O. Let $K\subset\mathbb{R}^{n}$ be a convex body, $n\geq 3$. If all
the $n-1$-dimensional sections of $K$ that divide the volume of $K$ in a given
ratio $\mu\neq 1$, have a centre of symmetry, then $K$ is an ellipsoid.

The following is a similar theorem, due to C. A. Rogers [8]:

Theorem R. Let $K\subset\mathbb{R}^{n}$ be a convex body, $n\geq 3$, and
$x\in\mathbb{R}^{n}$. If all the $2$-dimensional sections of $K$ through $x$
have a centre of symmetry, then $K$ has a centre of symmetry.

Rogers also conjectured that if $x$ was not the centre of $K$ then it is an
ellipsoid. This conjecture was first proved by P. W. Aitchison, C. M. Petty,
and C. A. Rogers [2], when $x\in\operatorname*{int}K$, and, the general case,
by D. G. Larman [4]. Later a simpler proof was given by L. Montejano and E.
Morales-Amaya in [6]. Such Theorem is known as the False Center Theorem.

In this note we prove, on the one hand, a result in $\mathbb{R}^{n}$ which is
a variant of the False Center Theorem when, instead of considering concurrent
planes, we consider planes tangent to a given sphere. It can be considered as
a progress about a problem due to J. A. Barker and D. Larman [3]. On the other
hand, we prove a result in $\mathbb{R}^{n}$ which is a variant of a Theorem of
L. Montejano which is a characterization of a ball in terms of concurrent
section of constant width [5].

In order to present our results we need the following definitions. Let
$K\subset\mathbb{R}^{n}$ be a convex body, $n\geq 3$. We say that $K$
satisfies the Barker-Larman Condition if there exists a sphere
$B\subset\text{int}K$ such that for every supporting hyperplane $\Pi$ of $B$
the section $\Pi\cap K$ is centrally symmetric. On the other hand, we say that
$K$ satisfies the Montejano Condition if there exists a sphere
$B\subset\text{int}K$ such that for every supporting hyperplane $\Pi$ of $B$
the section $\Pi\cap K$ is a body of constant width.

######  Theorem 1.

Let $K\subset\mathbb{R}^{n}$, $n\geq 3$, be an $O$-symmetric strictly convex
body. Suppose that $K$ satisfies the Barker-Larman condition for a ball $B$
which does not contain the $O$. Then $K$ is an ellipsoid.

######  Theorem 2.

Let $K\subset\mathbb{R}^{n}$, $n\geq 3$, be a convex body $O$-symmetric and
let $B\subset\operatorname*{int}K$ be a ball. Suppose that $O\notin B$ and,
for every supporting hyperplane $\Pi$ of $B$, the section $\Pi\cap K$ is a
body of constant width. Then $K$ is a ball.

##  2 Auxiliary result.

In this section we present a lemma which will be an important tool in the
proofs of Theorem 1 and 2 in dimension 3. The corresponding proof will be
given in the Appendix.

Let $M\subset\mathbb{R}^{2}$ be a $O$-symmetric convex body and let $B$ be a
circle such that $O\notin B$. We consider a system of coordinates with $O$ at
the origin. Given a fix point $w\in\operatorname*{bd}K$, we construct two
sequences $\\{a_{i}\\}_{i=1}^{\infty}$, $\\{b_{i}\\}_{i=1}^{\infty}$ in the
following manner. We make $a_{1}=w$ and for every point
$a_{i}\in\operatorname*{bd}M$ we take the point $b_{i}\in\operatorname*{bd}M$
such that the line $L(a_{i},b_{i})$ is a supporting line of $B$ and if $u_{i}$
is a unit vector with the property that the set of vector
$\\{b_{i}-a_{i},u_{i}\\}$ is a right frame of $\mathbb{R}^{2}$, then $B$ is
located in the half-space $a_{i}+\\{x\in\mathbb{R}^{2}:\langle
x,u_{i}\rangle>0\\}$ (See Fig. 1). On the other hand, for every
$b_{i}\in\operatorname*{bd}M$ we take the point
$a_{i+1}\in\operatorname*{bd}M$ such that the line $L(b_{i},a_{i+1})$ is a
supporting line of $-B$ and if $v_{i}$ is a unit vector with the property that
the set of vector $\\{a_{i+1}-b_{i},v_{i}\\}$ is a right frame of
$\mathbb{R}^{2}$, then $-B$ is located in the half-space
$b_{i+1}+\\{x\in\mathbb{R}^{2}:\langle x,v_{i}\rangle>0\\}$.

######  Lemma 1.

For the sequences $\\{a_{i}\\}_{i=1}^{\infty}$, $\\{b_{i}\\}_{i=1}^{\infty}$
there exists sub-sequences $\\{a_{i_{s}}\\}_{s=1}^{\infty}$,
$\\{b_{i_{s}}\\}_{s=1}^{\infty}$ which have the properties

$\displaystyle a_{i_{s}}\rightarrow a\textit{ and }b_{i_{s}}\rightarrow
b,\textit{ when }s\rightarrow\infty$ (1)

where $a,b\in M$ and $L(a,b)$ is a common supporting line of $B$ and $-B$ and
is passing through $O$.

Figure 1: The chords $a_{i}b_{i}$ converge to a common chord of $B$ and $-B$.

##  3 Proof of Theorem 1 in dimension 3.

Before to prove Theorem 1 we will prove a series of lemmas. We take a system
of coordinates such that $O$ is the origin. Let $\Pi$ be a plane through the
center $O$ of $K$. Let $\Gamma$ be a supporting plane of $B$ parallel to
$\Pi$. Since $K$ has centre at $O$ and, by hypothesis, $\Gamma\cap K$ has
centre, there exists a vector $u$ such that

$\displaystyle-(\Gamma\cap K)=u+(\Gamma\cap K).$ (2)

(Notice that a convex body $M\subset\mathbb{R}^{3}$ is centrally symmetric if
and only if $M$ and $-M$ are translated).

######  Lemma 2.

For every supporting plane $\Delta$ of $B$, parallel to $u$, the centre of the
section $\Delta\cap K$ is in $\Pi$, equivalently, the vector
$v\in\mathbb{R}^{3}$ with the property

$-(\Delta\cap K)=v+(\Delta\cap K).$

is parallel to $\Pi$.

###### Proof.

Let $\Delta$ be a supporting plane of $B$ parallel to $u$. We are going to
prove that the centre $z$ of $\Delta\cap K$ is in $\Pi$. By (2) it follows
that

$\displaystyle\Delta\cap(-\Gamma\cap K)=u+[\Delta\cap(\Gamma\cap K)].$ (3)

By virtue of (3) we conclude that the chords $\Delta\cap(-\Gamma\cap K)$ and
$\Delta\cap(\Gamma\cap K)$ of $\Delta\cap K$ are parallel and they have the
same length, from this and by strictly convexity of $\Delta\cap K$ it follows
that $z\in\Pi$. ∎

Let $\Phi_{u}:\mathbb{R}^{3}\rightarrow\Pi$ be the projection such that
$\operatorname*{ker}\Phi_{u}=\operatorname*{lin}\\{u\\}$. For
$u\in\mathbb{S}^{n-1}$, we denote by $S\partial(K,u)$ ($S\partial(K,L)$) the
shadow boundary of $K$ with respect to the vector $u$ (to the line $L$), i.e.,
the set of points $x$ in $\operatorname*{bd}K$ such that there exists a
supporting hyperplane of $K$ passing through $x$ and parallel to $u$ (parallel
to $L$).

######  Lemma 3.

If the planes $\Pi$, $\Gamma$ and the vector $u\in\mathbb{S}^{2}$ are such
that $O\notin\Phi_{u}(B)$, then the shadow boundary $S\partial(K,u)$ is
contained in a plane.

###### Proof.

We denote by $\xi(u)$ the shadow boundary $S\partial(K,u)$ of $K$ in the
direction $u$. Let $\Pi_{1},\Pi_{2}$ the two closed half-spaces determined by
$\Pi$. We claim that either $\xi(u)\subset\operatorname*{int}K_{1}$ or
$\xi(u)\subset\operatorname*{int}K_{2}$ are impossible, where
$K_{i}:=K\cap\Pi_{i}$, $i=1,2$. Since $\xi(u)$ is centrally symmetric with
centre at $O$, if $x\in\xi(u)$ and $x\in\operatorname*{int}K_{1}$, then
$-x\in\xi(u)$ and $-x\in\operatorname*{int}K_{2}$ (in virtue that
$\operatorname*{int}K_{2}=-\operatorname*{int}K_{1}$). Thus
$\xi(u)\cap\Pi\not=\emptyset$. Let $z\in\xi(u)\cap\Pi$. Let $L$ be a
supporting line of $K$ through $z$ parallel to $u$. For $K_{u}:=\Phi_{u}(K)$,
$B_{u}:=\Phi_{u}(B)$, $\bar{B}_{u}:=\Phi_{u}(-B)$ and $z=a_{1}$ we construct
the sequences $\\{a_{i}\\}$, $\\{b_{i}\\}$ as in the Lemma 1. By Lemma 1,
there exists sub-sequences $\\{a_{i_{s}}\\}$, $\\{b_{i_{s}}\\}$ such that

$\displaystyle a_{i_{s}}\rightarrow a\textrm{ and }b_{i_{s}}\rightarrow b,$
(4)

when $s\rightarrow\infty$, where $a,b\in\operatorname*{bd}K_{u}$ and $L(a,b)$
is a common tangent line of $B_{u}$ and $\bar{B}_{u}$ and through $O$. It is
clear that $\Delta_{s}:=\Phi_{u}^{-1}(L(a_{i_{s}},b_{i_{s}}))$ is a supporting
plane of $B$ and $\Sigma_{s}:=\Phi_{u}^{-1}(L(b_{i_{s}},a_{i_{s+1}}))$ is a
supporting plane of $-B$. By Lemma 2, the centre $O_{s}$ of the section
$\Delta_{s}\cap K$ is in $\Pi$ for all $s$ and the centre $P_{s}$ of the
section $\Sigma_{s}\cap K$ is in $\Pi$ for all $i$. Let
$\pi_{s}:\Delta_{s}\rightarrow\Delta_{s}$ and
$\rho_{s}:\Sigma_{s}\rightarrow\Sigma_{s}$ be the central symmetries with
respect to $O_{s}$ and $P_{s}$, respectively. Then
$b_{i_{s}}=\pi_{s}(a_{i_{s}})$ and $a_{{i_{s+1}}}=\rho_{s}(b_{i_{s}})$ (See
Fig. 2). It follows that $a_{i_{s}},b_{i_{s}}\in\xi(u)$ for all $s$. By (4),
it follows that

$\displaystyle a,b\in\xi(u).$ (5)

Figure 2: The shadow boundary $S\partial(K,u)$ is contained in a plane.

Let $x_{1}\in\xi(u)$, $x_{1}\not=z$. We are going to prove that $x_{1}\in\Pi$.
On the contrary, let assume that $x_{1}\notin\Pi$. Let $\delta>0$ the distance
from $x_{1}$ to $\Pi$. For $K_{u}:=\Phi_{u}(K)$, $B_{u}:=\Phi_{u}(B)$,
$\bar{B}_{u}:=\Phi_{u}(-B)$ and $a_{1}=\Phi_{u}(x_{1})$ we construct the
sequence $\\{a_{i}\\}$, $\\{b_{i}\\}$ as in the Lemma 1. By Lemma 1 there
exists sub sequences $\\{a_{i_{s}}\\}$, $\\{b_{i_{s}}\\}$ such that

$\displaystyle a_{i_{s}}\rightarrow a\textrm{ and }b_{i_{s}}\rightarrow b,$
(6)

when $s\rightarrow\infty$, where $a,b\in\operatorname*{bd}K_{u}$ and $L(a,b)$
is a common tangent line of $B_{u}$ and $\bar{B}_{u}$ and through $O$. We
define the plane $\Delta_{s},\Sigma_{s}$ and the maps $\pi_{s}$, $\rho_{s}$ as
before. We define the sequences $\\{y_{i_{s}}=\pi_{s}(x_{i_{s}})\\}$,
$\\{x_{i_{s+1}}:=\rho_{s}(y_{i_{s}})\\}\subset\operatorname*{bd}K$ and notice
that $a_{i_{s}}=\Phi_{u}(x_{i_{s}})$ and $b_{i_{s}}=\Phi_{u}(y_{i_{s}})$ (See
Fig. 2). By the Lemma 2, the distances between $y_{i_{s}}$ and $\Pi$ and
$x_{i_{s}}$ and $\Pi$ are equals to $\delta$ for all $s$. In virtue of (6) and
the compactness of $\operatorname*{bd}K$ there exists $x,y\in\xi(u)$ such that
$x_{i_{s}}\rightarrow x$ and $y_{i_{s}}\rightarrow y$, when
$s\rightarrow\infty$, and $\Phi_{u}(x)=a$ and $\Phi_{u}(y)=b$. Furthermore the
distances between $x$ and $\Pi$ and between $y$ and $\Pi$ are equal to
$\delta$. However this and (5) contradicts the strict convexity of $K$. ∎

Let $\Pi$ be a plane. We say the $\Pi$ strictly separates $B$ from $O$ if $B$
and $O$ are in different closed half-spaces generated by $\Pi$. We denote by
$\Omega$ the family of planes which strictly separate $B$ and $O$.

Let $\Sigma$ be a supporting plane of $B$ in $\Omega$. Let $\Pi$ be a plane
parallel to $\Sigma$ and with $O\in\Pi$. Let $\Sigma_{1},\Sigma_{2}$ be the
supporting planes of $K$ parallel to $\Pi$. We denote by $L_{\Sigma}$ the line
generated by the points $x_{i}:=\Sigma_{i}\cap\operatorname*{bd}K,i=1,2$.

Figure 3: The line $L_{\Sigma}$ is affine axis of symmetry of $K$.

######  Lemma 4.

For all $\Sigma\in\Omega$ the line $L_{\Sigma}$ is affine axis of symmetry of
$K$ and all the sections of $K$ parallel to $\Sigma$ are similar and similarly
situated.

###### Proof.

By virtue that $\Sigma\cap K$ has centre, the section $-\Sigma\cap K$ has
centre. Furthermore, there exists $v\in\mathbb{R}^{3}$ such that $-\Sigma\cap
K=v+(\Sigma\cap K)$. By Lemma 2, for every supporting plane $\Delta$ of $B$,
parallel to $v$, the centre of the section $\Delta\cap K$ is in $\Pi$,
equivalently, the vector $w\in\mathbb{R}^{3}$ with the property $-(\Delta\cap
K)=w+(\Delta\cap K)$is parallel to $\Pi$ (See Fig. 3). Consequently, by the
choice of $\Sigma$, $O\notin\Phi_{w}(B)$. Thus, by Lemma 3, $S\partial(K,w)$
is contained in a plane. It follows that every shadow boundary
$S\partial(K,w)$ of $K$ corresponding a direction $w$, parallel to $\Pi$, is
given as the section $H\cap K$, where $H$ is a plane containing $L_{\Sigma}$.
Therefore all the sections of $K$ parallel to $\Sigma$ are centrally
symmetric, with centre at $L_{\Sigma}$, similar and similarly situated (A
proof of the later can be found in [4]). ∎

The following notion will be important in the proof of the next lemma.
According to [9], the convex floating body $K_{\delta}$ of a convex body $K$
in $\mathbb{R}^{n}$ is the intersection of all halfspaces whose hyperplanes
cut off a set of volume $\delta$ from $K$, for our purposes it is enough to
take $\delta$ such that it satisfies the inequalities
$0<\delta<\frac{\textrm{Vol }(K)}{2}$.

######  Lemma 5.

$K$ is affine equivalent to a body of revolution.

###### Proof.

We denote by $\lambda$ the center of $B$. By a continuity argument, we
conclude the existence of a supporting plane $\Gamma$ of $B$ in $\Omega$ such
that

$\displaystyle-\Gamma\cap K=-a\lambda+(\Gamma\cap K)$ (7)

for a real number $a$, $0<a<1$. By Lemma 4, the line $L_{\Gamma}$ is affine
axis of symmetry of $K$ and all the sections of $K$, parallel to $\Gamma$, are
similar, similarly situated and with centres at $L_{\Gamma}$. Let $\Pi$ be a
plane parallel to $\Gamma$ and with $O\in\Pi$. In order to complete the proof
of Lemma 5, we are going to prove that the section
$\Pi\cap\operatorname*{bd}K$ is an ellipse. We consider an affine
transformation $A:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ such that
$A(\Pi)=\Pi$ and $A(\Pi)\perp A(L_{\Sigma})$, i.e. $A(L_{\Sigma})$ is an axis
of symmetry of $A(K)$. We will use the same notation for the geometric objects
which are involve in the Lemma 7 after applied them the transformation $A$,
that is, we will denote by $K$ the set $A(K)$, by $L_{\Sigma}$ the set
$A(L_{\Sigma})$ and so on.

We claim that the relation

$\displaystyle S\partial(K,L_{\Gamma})=\Pi\cap\operatorname*{bd}K$ (8)

holds. Since $K$ is $O$-symmetric and strictly convex, it is clear that,
between all the sections of $K$ parallel to $\Gamma$, the section
$\Pi\cap\operatorname*{bd}K$ is which has bigger area. Thus (8) follows from
this and from the fact that all the sections of $K$, parallel to $\Gamma$, are
similar, similarly situated and with centres at $L_{\Gamma}$.

By (8) we conclude that the projection $K_{\lambda}:=\Phi_{\lambda}(K)$ is
equal to the section $\Pi\cap\operatorname*{bd}K$. Now we are going to prove
that the projection $K_{\lambda}$ is an ellipse.

Notice that

  * $(*)$

If a convex figure $K\subset\mathbb{R}^{2}$ is $O$-symmetric and it has a line
of symmetry $\mathcal{L}$ passing through $O$, then the line $\mathcal{M}$
perpendicular to $\mathcal{L}$, passing through $O$, is a line of symmetry of
$M$.

We claim the $\Pi$ is a plane of symmetry of $K$. Since $L_{\Gamma}$ is axis
of rotation of $K$, for all plane $\Delta$ containing $L_{\Gamma}$, the
section $\Delta\cap K$ is $O$-symmetric and it has the $L_{\Gamma}$ as line of
symmetry. Then, by $(*)$ the line perpendicular to $L_{\Gamma}$ passing
through $O$ is a line of symmetry. Varying $\Delta$,
$L_{\Gamma}\subset\Delta$, it follows that $\Pi$ is plane of symmetry of $K$.

Now we are going to show that for every plane $\Sigma$, parallel to
$L_{\Gamma}$ and tangent to $B$, the section $\Sigma\cap K$ has a line of
symmetry $l_{\Sigma}$ parallel to $L_{\Gamma}$. By hypothesis $\Sigma\cap K$
is centrally symmetric, say with centre at $c_{\Sigma}$. On the one hand, by
Lemma 2, $c_{\Sigma}\in\Pi$. Hence $c_{\Sigma}\in\Sigma\cap\Pi$. On the other
hand, by virtue that $\Pi$ is plane of symmetry of $K$ and $\Sigma$ is
parallel to $L_{\Gamma}$, the section $\Sigma\cap K$ has the line
$\Sigma\cap\Pi$ as line of symmetry. Thus, by $(*)$, $\Sigma\cap\Pi$ has line
of symmetry $l_{\Sigma}$ parallel to $L_{\Gamma}$.

Let $\Sigma$ be a plane parallel to $L_{\Gamma}$ and tangent to $B$. We denote
by $\Sigma^{+}$ and $\Sigma^{-}$ the half-spaces defined by $\Sigma$, choosing
the notation such that $O\in\Sigma^{-}$. Let $\Pi_{1}$ be a plane parallel to
$\Pi$ and such that

$\displaystyle\Pi_{1}\cap\operatorname*{int}(\Sigma\cap K)\not=\emptyset.$ (9)

We denote by $W_{\Sigma}$ the plane containing the parallel lines $l_{\Sigma}$
and $L_{\Gamma}$ and by $a,b$ the extreme points of the chord
$\Pi_{1}\cap(\Sigma\cap K)$. Since $l_{\Sigma}$ is line of symmetry of
$\Sigma\cap K$ the chord $ab$ has it mid-point $c$ at $l_{\Sigma}\subset
W_{\Sigma}$. It follows that the chord with extreme points
$a_{\lambda}:=\phi_{\lambda}(a)$, $b_{\lambda}:=\phi_{\lambda}(b)$ has its
mid-point $c_{\lambda}:=\phi_{\lambda}(c)$ at $m:=W_{\Sigma}\cap\Pi$ and it is
tangent to the ellipse $B_{\lambda}:=\phi_{\lambda}(B)$ (See Fig. 4). By
virtue that $K_{\lambda}$ and $\Pi_{1}\cap K$ are homothetic, then
$K_{\lambda}$ and $K_{\lambda}^{1}:=\phi_{\lambda}(\Pi_{1}\cap K)$ are
homothetic with centre of homothety $O$. Let $h:\Pi\rightarrow\Pi$ be the
homothecy, with centre of homothety $O$, such that
$h(K_{\lambda}^{1}))=K_{\lambda}$. We conclude that the chord
$h(a_{\lambda})h(b_{\lambda})$ of $K_{\lambda}$ has its mid-point at $m$.
Varying $\Pi$, always parallel to $\Pi$ and such that the relation (9) holds,
we get to the following conclusions:

  * 1)

All the chords of $K_{\lambda}$, parallel to the line $\Pi\cap\Sigma$ and
contained in the half-plane $\Pi\cap\Sigma^{+}$, have its mid-point at the
line $m$ (See Fig. 4).

  * 2)

There exist a supporting line $n$ of $K_{\lambda}$ at the point $\tau$ in
$m\cap K_{\lambda}$, with $\tau\in\Pi\cap\Sigma^{+}$, parallel to the line
$\Pi\cap\Sigma$.

  * 3)

For all plane $\Pi_{1}$, parallel to $\Pi$ and close enough to $\Pi$, the
convex figure $K_{\lambda}^{1}$ is the floating body of $K_{\lambda}$.

By 3) and Theorem 1 of [9] it follows that $K_{\lambda}$ is an ellipse.

Figure 4: The convex figure $K_{\lambda}^{1}$ is the floating body of
$K_{\lambda}$.

∎

###  3.1 Prove of Theorem 1.

We denote by $A(3)$ and $O(3)$ the sets of all the affine and orthogonal
transformations from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$, respectively. By
virtue of Lemma 5 we can find $A\in A(3)$ such that $G:=A(L_{\Gamma})$ is an
axis of revolution of $\bar{K}:=A(K)$. By Lemma 4, for $\Sigma\in\Omega$, the
line $L_{\Sigma}$ is an affine axis of symmetry of $K$. Thus the line
$J_{\Sigma}:=A(L_{\Sigma})$ is an affine axis of symmetry of $\bar{K}$. Let

$\Lambda_{\Sigma}=\\{T(J_{\Sigma}):T\in O(3)\textrm{ such that }T(G)=G\\}.$

For each line $Q\in\Lambda_{\Sigma}$, let $A_{Q}\in A(3)$ such that
$\bar{Q}=A_{Q}(Q)$ is an axis of symmetry of $\bar{K}_{Q}:=A_{Q}(\bar{K})$
and, consequently, $A_{Q}(G)$ is an affine axis of revolution of
$\bar{K}_{Q}$. Let

$\mathrm{\cal{A}}_{\Sigma}=\\{A_{Q}\in A(3):Q\in\Lambda_{\Sigma}\\}.$

For each $Q\in\Lambda_{\Sigma}$, let $R_{\bar{Q}}\in O(3)$ be the rotation
with axis $\bar{Q}$ by an angle $\pi$. Thus
$R_{\bar{Q}}(\bar{K}_{Q})=\bar{K}_{Q}$. Let

$\mathrm{\cal{B}}_{\Sigma}=\\{R_{\bar{Q}}\in
O(3):Q\in\Lambda_{\Sigma}\\}\textrm{ }\textrm{ and }\textrm{
}\mathrm{\cal{C}}_{\Sigma}=\\{A_{Q}^{-1}(R_{\bar{Q}}(A_{Q}(G))):Q\in\Lambda_{\Sigma}\\}$

We can illustrates these definitions with the following diagram:

$K\stackrel{{\scriptstyle A}}{{\longrightarrow}}\bar{K}\stackrel{{\scriptstyle
A_{Q}}}{{\longrightarrow}}\bar{K}_{Q}\stackrel{{\scriptstyle
R_{\bar{Q}}}}{{\longrightarrow}}\bar{K}_{Q}\stackrel{{\scriptstyle
A^{-1}_{Q}}}{{\longrightarrow}}\bar{K}\stackrel{{\scriptstyle
A^{-1}}}{{\longrightarrow}}K$

Since

$A_{Q}^{-1}(R_{\bar{Q}}(A_{Q}(\bar{K})))=A_{Q}^{-1}(R_{\bar{Q}}(\bar{K}_{Q}))=A_{Q}^{-1}(\bar{K}_{Q})=A_{Q}^{-1}(A_{Q}(\bar{K}))=\bar{K}$

every element of $\mathrm{\cal{C}}_{\Sigma}$ is an affine axis of revolution
of $\bar{K}$. Consequently, for each $W\in\mathrm{\cal{C}}_{\Sigma}$,
$A^{-1}(W)$ is an affine axis of revolution of $K$.

Suppose that there exists $\Sigma_{1}\in\Omega$ such that, for all
$Q\in\Lambda_{\Sigma_{1}}$, $A_{Q}=\textrm{id}$ and $\Lambda_{\Sigma_{1}}$ is
the equator of $\bar{K}$ ($\mathrm{\cal{C}}_{\Sigma_{1}}$ consist just of the
element $G$). Thus, on the one hand, $A_{Q}(G)=\textrm{id}(G)=G$, and, on the
other hand, by virtue that $G$ and $Q$ are orthogonal, $R_{\bar{Q}}(G)=G$.
Hence

$A_{Q}^{-1}(R_{\bar{Q}}(A_{Q}(G)))=A_{Q}^{-1}(R_{\bar{Q}}(\textrm{id}(G)))=A_{Q}^{-1}(R_{\bar{Q}}(G))=A_{Q}^{-1}(G)=G.$

Let $\Sigma_{2}\in\Omega$, $\Sigma_{1}\not=\Sigma_{2}$. Suppose that
$\Sigma_{2}$ is such that $J_{\Sigma_{2}}:=A(L_{\Sigma_{2}})$ is an axis of
symmetry of $\bar{K}$, i.e., $A_{\Sigma_{2}}=\textrm{id}$. Then $\bar{K}$ is
an sphere by virtue that it has two different axis of revolution, namely, $G$
and $R_{\bar{\Sigma}_{2}}(G)$. Consequently, $K$ is an ellipsoid.

Due to what has been seen, we can assume that
$J_{\Sigma_{2}}:=A(L_{\Sigma_{2}})$ is an axis of affine symmetry, i.e.,
$A_{\Sigma_{2}}\not=\textrm{id}$. We denote by $\tau$ the colection of planes
containing $G$. For each $\Pi\in\tau$, we will call the set $\Pi\cap\bar{K}$ a
meridian of $\bar{K}$. For each $Q\in\Lambda_{\Sigma_{2}}$, we denote by
$\zeta_{Q}$ the meridian passing through $Q$. Since for every
$Q\in\Lambda_{\Sigma_{2}}$, $A_{Q}(G)$ and $R_{\bar{Q}}(A_{Q}(G))$ are affine
axis of revolution of $\bar{K}_{Q}$, it follows that each element in
$\mathrm{\cal{C}}_{\Sigma_{2}}$ is an affine axis of revolution of $\bar{K}$.
Notice that for $Q\in\Lambda_{\Sigma_{2}}$,
$A_{Q}^{-1}(R_{\bar{Q}}(A_{Q}(G)))\subset\zeta_{Q}$. Consequently, $\bar{K}$
has an infinite number of affine axis of revolution. By Theorem 1 of [1], the
only convex body with an infinite number of affine axis of revolution is the
ellipsoid. Hence $\bar{K}$ is an ellipsoid. Therefore $K$ is an ellipsoid.

If $\mathrm{\cal{C}}_{\Sigma_{1}}$ no consist of just one element, then $K$
has an infinite number of affine axis of revolution (see the previous
paragraph) and, again by Theorem 1 of [1], it follows that $K$ is an
ellipsoid.

##  4 Proof of Theorem 2 in dimension 3.

We take a system of coordinates such $O$ is the origin. We denote by
$\Omega_{O}$ the solid cone $\\{\alpha x:x\in B,\alpha\in\mathbb{R}\\}$, by
$u$ the centre of $B$, by $L$ the line determined by the centres of $B$ and
$-B$.

######  Lemma 6.

For every $v\in\mathbb{S}^{2}\cap u^{\perp}$ the relation

$\displaystyle v^{\perp}\cap\operatorname*{bd}K=S\partial(K,v)$ (10)

holds.

###### Proof.

We observe that since every figure of constant width is strictly convex the
body $K$ is is strictly convex. We denote by $\xi(v)$ the shadow boundary
$S\partial(K,v)$ of $K$ in the direction $v.$ Let $\Pi_{1},\Pi_{2}$ the two
closed half-spaces determined by $v^{\perp}$. We claim that either
$\xi(v)\subset\operatorname*{int}K_{1}$ or
$\xi(v)\subset\operatorname*{int}K_{2}$ are impossible, where
$K_{i}:=\operatorname*{bd}K\cap\Pi_{i}$, $i=1,2$. Since $\xi(v)$ is centrally
symmetric with center at $O$, if $x\in\xi(v)$ and
$x\in\operatorname*{int}K_{1}$, then $-x\in\xi(v)$ and
$-x\in\operatorname*{int}K_{2}$ (in virtue that
$\operatorname*{int}K_{2}=-\operatorname*{int}K_{1}$). Thus $\xi(v)\cap
v^{\perp}\not=\emptyset$. Let $z\in\xi(v)\cap v^{\perp}$. Let $T$ be a
supporting line of $K$ through $z$ parallel to $v$. Let
$\phi:\mathbb{R}^{3}\rightarrow v^{\perp}$ be the projection such that
$\operatorname*{ker}\phi=\operatorname*{lin}\\{v\\}$. For $K_{v}:=\phi(K)$,
$B_{v}:=\phi(B)$, $\bar{B}_{v}:=\phi(-B)$ and $z=a_{1}$ we construct the
sequences $\\{a_{i}\\}_{i}^{\infty}$, $\\{b_{i}\\}_{i}^{\infty}$ as in the
Lemma 1. By Lemma 1, $a_{i}\rightarrow a$ and $b_{i}\rightarrow b$, where
$a,b\in\operatorname*{bd}K_{v}$ and $L(a,b)$ is a common tangent line of
$B_{v}$ and $\bar{B}_{v}$ and through $O$. It is clear that
$\Delta_{i}:=\phi^{-1}(L(a_{i},b_{i}))$ is a supporting plane of $B$ and
$\Sigma_{i}:=\phi^{-1}(L(b_{i},a_{i+1}))$ is a supporting plane of $-B$.
Notice that the chord $[a_{i},b_{i}]$ is a bi-normal of $\Delta_{i}\cap K$ and
the chord $[b_{i},a_{i+1}]$ is a bi-normal of $\Sigma_{i}\cap K$. Thus, since
$z=a_{1}$, in each point $a_{i}$, $b_{i}$ there exists supporting lines
parallel to $T$. It follows that $a_{i},b_{i}\in\xi(v)$ for all $i$. Hence

$\displaystyle a,b\in\xi(v).$ (11)

Let $x_{1}\in\xi(v)$, $x_{1}\not=z$. We are going to prove that $x_{1}\in
v^{\perp}$. On the contrary, let assume that $x_{1}\notin v^{\perp}$. Let
$\delta>0$ be the distance from $x_{1}$ to $v^{\perp}$. For $K_{v}:=\phi(K)$,
$B_{v}:=\phi(B)$, $\bar{B}_{v}:=\phi(-B)$ and $a_{1}=\phi(x_{1})$ we construct
the sequence $\\{a_{i}\\}_{i=1}^{\infty}$, $\\{b_{i}\\}_{i=1}^{\infty}$ as in
the Lemma 1. By Lemma 1, $a_{i}\rightarrow a$ and $b_{i}\rightarrow b$, when
$i\rightarrow\infty$, where $a,b\in\operatorname*{bd}K_{v}$ and $L(a,b)$ is a
common tangent line of $B_{v}$ and $\bar{B}_{v}$ and through $O$.

We define the sequence $\\{x_{i}\\}$, $\\{y_{i}\\}\subset\operatorname*{bd}K$
such that $a_{i}=\phi(x_{i})$ and $b_{i}=\phi(y_{i})$ (by virtue of the
strictly convexity of $K$ such sequence are well defined). The planes
$\Delta_{i}:=\phi^{-1}(L(a_{i},b_{i}))$ and
$\Sigma_{i}:=\phi^{-1}(L(b_{i},a_{i+1}))$ are a supporting plane of $B$ and of
$-B$, respectively. Notice that the chord $[x_{i},y_{i}]$ is a bi-normal of
$\Delta_{i}\cap K$ and the chord $[y_{i},x_{i+1}]$ is a bi-normal of
$\Sigma_{i}\cap K$ (there are supporting lines of $K$ parallel to $T$ at the
points $x_{i},y_{i},x_{i+1}$ hence the chords $[x_{i},y_{i}]$,
$[y_{i},x_{i+1}]$ are diametral chords of $\Delta_{i}\cap K$ and
$\Sigma_{i}\cap K$, respectively, and, consequently they are bi-normal). Thus
the chords $[x_{i},y_{i}]$ and $[y_{i},x_{i+1}]$ are parallel to $v^{\perp}$.
Therefore the distances between $y_{i}$ and $v^{\perp}$ and $x_{i}$ and
$v^{\perp}$ are equals to $\delta$ for all $i$. Since $a_{i}\rightarrow a$ and
$b_{i}\rightarrow b$, when $i\rightarrow\infty$, there exists $x,y\in\xi(v)$
such that $x_{i}\rightarrow x$ and $y_{i}\rightarrow y$, when
$i\rightarrow\infty$, and $\phi(x)=a$ and $\phi(y)=b$. Furthermore the
distances between $x$ and $v^{\perp}$ and between $y$ and $v^{\perp}$ are
equal to $\delta$. However this and (11) contradicts the strict convexity of
$K$. ∎

Proof of Theorem 2. We observe that for every supporting plane $\Gamma$ of
$B$, passing through $O$, the section $\Gamma\cap K$ is a disc with centre at
$O$, since it is of constant width and centrally symmetric. Let $R$ be the
radius of the disc $\Gamma\cap K$ and let $G$ be a ball of radius $R$ and with
centre at $O$. We denote by $K_{\Omega}$ the set
$(\mathbb{R}^{3}\backslash\Omega_{O})\cap\operatorname*{bd}K$. We are going to
prove $K_{\Omega}\subset G$. Let $x\in K_{\Omega}\backslash\Gamma$. Let $\Pi$
be a supporting plane of $B$ containing the line $L(O,x)$. For one hand,
$\Gamma\cap K$ and $\Pi\cap K$ are discs and, on the other hand, the chord
$\Gamma\cap\Pi\cap K$ is a common diameter of them (both discs have center at
$O$). Thus the radius of $\Pi\cap K$ is equal to $R$. Hence $x\in G$.

By Lemma 6 the sections of $K$ with planes parallel to $u^{\perp}$ are similar
and since $u^{\perp}\cap K$ is a disc (notice that $u^{\perp}\cap K\subset
G$), all the sections of $K$ with planes parallel to $u^{\perp}$ are discs.
Thus $K$ is a body of revolution with axis the line $L$.

Finally, if we take a point $w\in\operatorname*{bd}K\backslash K_{\Omega}$,
$w\notin L$, and we repeat the previous argument we can prove that $K$ is a
body of revolution con respect to the line $L(O,w)$ with $L(O,w)\not=L$.
Consequently, $K$ is a body of revolution with two different axis of symmetry.
Hence it is a ball.

##  5 Proof of Theorem 1 in dimension $>3$.

We will assume that Theorem 1 holds in dimension $n-1$ and we will prove that
it holds in dimension $n$, $n\geq 4$. We take a system of coordinates such $O$
is the origin. We denote by $\phi_{u}:\mathbb{R}^{n}\rightarrow u^{\perp}$ the
orthogonal projection con respect to the direction $u\in\mathbb{S}^{n-1}$. Let
$\Pi$ be a hyperplane which separates $B$ from $o$.

######  Lemma 7.

For every vector $u\in\mathbb{S}^{n-1}$ parallel to $\Pi$ the projection
$\phi_{u}(K)$ is an $(n-1)$-ellipsoid.

###### Proof.

First notice, by the choice of $\Pi$, that $O\notin\phi_{u}(B)$. Since every
orthogonal projection of a $O$-symmetric body is a $O$-symmetric body in
dimension $n-1$, the $O$-symmetric body $\phi_{u}(K)$ satisfies the Barker-
Larman condition with respect to sphere $\phi_{u}(B)$ (every $(n-2)$-section
of $\phi_{u}(K)$ given by supporting $(n-2)$-plane of $\phi_{u}(B)$ can be
considered as the projection of one $(n-1)$-section of $K$ given by an
hyperplane tangent to $B$ and parallel to $u$). By virtue of the induction
hypothesis, $\phi_{u}(K)$ is an $(n-1)$-ellipsoid. ∎

Let $\Pi$ be a hyperplane which separates $B$ from $o$. Let $\Pi_{1},\Pi_{2}$
be supporting hyperplanes of $K$ at the points
$x_{1},x_{2}\subset\operatorname*{bd}K$, respectively. Let $\Gamma$ be a
hyperplane parallel to $\Pi$, $O\in\Gamma$. We denote by $L$ the line
generated by the points $x_{1}$, $x_{2}$.

######  Lemma 8.

$\Gamma$ The relation

$\displaystyle\Gamma\cap\operatorname*{bd}K=S\partial(K,L)$ (12)

holds.

###### Proof.

WLG we can assume that $L\perp\Pi$. Let $z\in\Gamma\cap\operatorname*{bd}K$
and let $\Delta$ be a supporting hyperplane of $K$ at $z$. Let $w$ be a unit
vector parallel to $\Delta\cap\Pi$. By Lemma 7, $\phi_{w}(K)$ is an
ellipsoide. Since $\phi_{w}(K)$ is an ellipsoide and by virtue that
$\phi_{w}(L)=L$, and $\phi_{w}(\Gamma)$ is parallel to the supporting
$(n-2)$-planes $\phi_{w}(\Pi_{1})$, $\phi_{w}(\Pi_{2})$ of $\phi_{w}(K)$ and
it is passing through $o$, it follows that

$\displaystyle S\partial(\phi_{w}(K),L)=\phi_{w}(\Gamma)\cap\phi_{w}(K)$ (13)

Thus there exists a unique supporting $(n-2)$-plane $\Sigma$ of $\phi_{w}(K)$
at $\phi_{w}(z)$ parallel to $L$. Consequently, $\Delta=\phi_{w}^{-1}(\Sigma)$
is parallel to $L$. Therefore $z\in S\partial(K,L)$. Hence
$\Gamma\cap\operatorname*{bd}K\subset S\partial(K,L)$.

On the other hand, let $z\in S\partial(K,L)$. Let $\Delta$ be a supporting
hyperplane of $K$ at $z$ parallel to $L$. Let $w$ be a unit vector parallel to
$\Delta\cap\Pi$. By Lemma 7, $\phi_{w}(K)$ is an elipsoide. By virtue that
$\phi_{w}(\Delta)$ is parallel to $L$ and since the relation (13) holds, it
follows that $\phi_{w}(z)\in\phi_{w}(\Gamma)\cap\phi_{w}(K)$. Thus
$z\in\Gamma\cap\operatorname*{bd}K$, that is,
$S\partial(K,L)\subset\Gamma\cap\operatorname*{bd}K$ ∎

######  Lemma 9.

There exists an affine transformation
$A:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $A(\Gamma)=\Gamma$,
$A(L)\perp\Gamma$ and the ellipsoids
$\\{\phi_{u}(A(K)):u\in\mathbb{S}^{n-1},u\textrm{ parallel to }\Gamma\\}$ are
congruent ellipsoids of revolution with axis $A(L)$, that is, $A(K)$ is an
ellipsoid of revolution with axis $A(L)$.

###### Proof.

For all $u\in\mathbb{S}^{n-1}$, $u$ parallel to $\Gamma$, the next series of
equals holds

$\displaystyle\phi_{u}(\Gamma\cap\operatorname*{bd}K)=\phi_{u}(S\partial(K,L))=S\partial(\phi_{u}(K),L)=\phi_{u}(\Gamma)\cap\phi_{u}(K).$
(14)

The first equality is by (12) of Lemma 8, the second follows immediately from
the definition of shadow boundary and the last is by (13). It follows that the
$(n-1)$-section $\Gamma\cap\operatorname*{bd}K$ is an ellipsoide because all
its orthogonal projection are $(n-2)$-ellipsoides (notice that, by virtue of
Lemma 7, $\phi_{u}(\Gamma)\cap\phi_{u}(K)$ is a $(n-2)$-ellipsoide).

Now we choose an affine transformation
$A:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $A(\Gamma)=\Gamma$,
$A(L)\perp\Gamma$, and $A(\Gamma\cap\operatorname*{bd}K)$ is a sphere of
radius $R$. Thus, for all $u\in\mathbb{S}^{n-1}$, $u$ parallel to $\Gamma$,
$\phi_{u}(A(K))$ is an ellipsoid of revolution with axis $A(L)$ and such that
$A(L)\cap\phi_{u}(A(K))=\\{A(x_{1}),A(x_{2})\\}$ and with the maximum radius
of the spheres perpendicular to $A(L)$ equal to $R$. ∎

##  6 Proof of Theorem 2 in dimension $>3$.

We will assume that Theorem 2 holds in dimension $n-1$, and we will prove that
it holds in dimension $n$, $n\geq 4$. We take a system of coordinates such $O$
is the origin. We denote by $\phi_{u}:\mathbb{R}^{n}\rightarrow u^{\perp}$ the
orthogonal projection con respect to the direction $u\in\mathbb{S}^{n-1}$. Let
$\Pi$ be a hyperplane which separates $B$ from $o$.

######  Lemma 10.

For every vector $u\in\mathbb{S}^{n-1}$ parallel to $\Pi$ the projection
$\phi_{u}(K)$ is an $(n-1)$-ball.

###### Proof.

By the choice of $\Pi$ it follows that $O\notin\phi_{u}(B)$. Since every
orthogonal projection of a body of constant width is a body of constant width
in dimension $n-1$, the $O$-symmetric body $\phi_{u}(K)$ satisfies the
Montejano’s condition with respect to sphere $\phi_{u}(B)$ (every
$(n-2)$-section of $\phi_{u}(K)$ given by supporting $(n-2)$-plane of
$\phi_{u}(B)$ can be considered as the projection of one $(n-1)$-section of
$K$ given by an hyperplane tangent to $B$ and parallel to $u$). By virtue of
the induction hypothesis, $\phi_{u}(K)$ is an $(n-1)$-ball. ∎

Let $\Pi_{1},\Pi_{2}$ be supporting hyperplanes of $K$, parallel to $\Pi$, at
the points $\\{x_{1},x_{2}\\}\subset\operatorname*{bd}K$, respectively. We
denote by $L$ the line generated by the points $x_{1}$, $x_{2}$. By virtue of
Lemma 10, the chord $x_{1}x_{2}$ of $K$ is bi-normal chord of $K$. It follows
that all the $(n-1)$-balls $\phi_{u}(K)$, $u\in\mathbb{S}^{n-1}$ parallel to
$\Pi$, have the same radius. Let $\Phi$ be the $n$-ball with diameter
$x_{1}x_{2}$. We claim that $K=\Phi$. On the contrary, suppose that there is a
point $x\in\Phi$ and such that $x\notin K$. Let $\Gamma$ be a hyperplane which
separates $x$ and $K$. Let $u\in\mathbb{S}^{n-1}$ parallel to $\Pi\cap\Gamma$.
Then, on the one hand, $\Gamma\cap u^{\perp}$ separates $\phi_{u}(K)$ and
$\phi_{u}(x)$, on the other hand, since $\phi_{u}(x)\in\phi_{u}(\Phi)$ and
$\phi_{u}(\Phi)=\phi_{u}(K)$ (both are balls with diameter $x_{1}x_{2}$), it
follows that $\phi_{u}(x)\in\phi_{u}(K)$. This contradiction shows that
$\Phi\subset K$. The opposite inclusion can be seen in analogous way. Thus
$K=\Phi$.

Acknowledment. The author is very thankful to Jesús Jeronimo Castro for very
useful discussions.

##  7 Appendix.

###  7.1 Proof of Lemma 1

We will prove the lemma by contradiction. In virtue of the compactness of $M$
there exists a point $a\in\operatorname*{bd}M$ and a sub-sequence
$\\{a_{i_{s}}\\}$ of $\\{a_{i}\\}$ such that $a_{i_{s}}\rightarrow a$,
$s\rightarrow\infty$. Let us assume that there exists a point
$b\in\operatorname*{bd}M$ for which the following two properties holds: 1) the
sub-sequence $\\{b_{i_{s}}\\}$ of $\\{b_{i}\\}$ is such that
$b_{i_{s}}\rightarrow b$, $s\rightarrow\infty$ and 2) $L(a,b)$ is not a common
supporting line of $B$ and $-B$ passing through $O$. In particular, suppose
that $L(a,b)$ is not a supporting line of $B$. Let
$W_{1},W_{2}\subset\mathbb{R}^{2}$ be two lines parallel to $L(a,b)$ and whose
distance to $L(a,b)$ are equal to $\epsilon$, $0<\epsilon<1$. We denote by $D$
the band determined by $W_{1},W_{2}$. Since $a_{i_{s}}\rightarrow a$,
$s\rightarrow\infty$, there exists $N_{1}\in\mathbb{N}$ such that
$|a_{i_{s}}-a|<\epsilon$ for $s>N_{1}$. On the other hand, in virtue that
$b_{i_{s}}\rightarrow b$, $s\rightarrow\infty$, there exists
$N_{2}\in\mathbb{N}$ such that $|b_{i_{s}}-b|<\epsilon$ for $s>N_{2}$. Let
$N:=\max\\{N_{1},N_{2}\\}$. We take $s_{0}>N$. Then

$\displaystyle|a_{i_{s_{0}}}-a|<\epsilon\textrm{ and
}|b_{i_{s_{0}}}-b|<\epsilon,$ (15)

i.e., the chord $[a_{i_{s_{0}}},b_{i_{s_{0}}}]$ is contained in $D$

On the other hand, given $x\in\operatorname*{bd}M$ we denote by $l(x),m(x)$
the supporting lines of $B$ passing through $x$ and by $y(x),z(x)$ the
intersections of $l(x)$ and $m(x)$ with $\operatorname*{bd}M$, respectively,
$y(x)\not=x\not=z(x)$. Since $L(a,b)$ is not a supporting line of $B$ we can
see that if $\epsilon>0$ is small enough, for each $x\in\operatorname*{bd}M$
such that $|x-b|<\epsilon$ the chords $[x,y(x)]$, $[x,z(x)]$ does not belong
to $D$. Thus $|a_{i_{s_{0}}}-a|>\epsilon$ which contradict (15).

If the sub-sequence $\\{b_{i_{s}}\\}$ does not converge, a sub-sequence of it
$\\{b_{i_{s_{r}}}\\}$ will converge to a point $b\in\operatorname*{bd}M$. Now
we assume again that $L(a,b)$ is not a common supporting line of $B$ and $-B$
passing through $O$. Furthermore we can assume that $L(a,b)$ is not a
supporting line of $B$. Finally we repeat the previous argument for the
sequences $\\{a_{i_{s_{r}}}\\}$, $\\{b_{i_{s_{r}}}\\}$ and we will get a
contradiction again.

The other cases can be considered in analogous way.

## References

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  * [2] P. W. Aitchison, C. M. Petty, and C. A. Rogers, _A convex body with a false centre is an ellipsoid._ Mathematika 18 (1971), $50-59$.
  * [3] J. A. Barker, and D. G. Larman, _Determination of convex bodies by certain sets of sectional volumes._ Discrete Math. 241 (2001), $79-96$.
  * [4] D.G. Larman: _A note on the false centre problem_ , Mathematika 21 (1974), $216-217$.
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  * [6] L. Montejano, and E. Morales-Amaya, _Variations of classic characterizations of ellipsoids and a short proof of the false centre theorem._ Mathematika 54 (2007), no. 1-2, $35-40$.
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# Variational Autoencoding of Dental Point Clouds

Johan Ziruo Ye  Thomas Ørkild  Peter Lempel Søndergaard  Søren Hauberg

###### Abstract

Digital dentistry has made significant advancements, yet numerous challenges
remain. This paper introduces the _FDI 16_ dataset, an extensive collection of
tooth meshes and point clouds. Additionally, we present a novel approach:
_Variational FoldingNet (VF-Net)_ , a fully probabilistic variational
autoencoder designed for point clouds. Notably, prior latent variable models
for point clouds lack a one-to-one correspondence between input and output
points. Instead, they rely on optimizing Chamfer distances, a metric that
lacks a normalized distributional counterpart, rendering it unsuitable for
probabilistic modeling. We replace the explicit minimization of Chamfer
distances with a suitable encoder, increasing computational efficiency while
simplifying the probabilistic extension. This allows for straightforward
application in various tasks, including mesh generation, shape completion, and
representation learning. Empirically, we provide evidence of lower
reconstruction error in dental reconstruction and interpolation, showcasing
state-of-the-art performance in dental sample generation while identifying
valuable latent representations. 111Code at https://github.com/JohanYe/VF-Net

Machine Learning, ICML

##  1 Introduction

Recent advancements and widespread adoption of intraoral scanners in dentistry
have made micrometer-resolution 3D models readily available. Consequently, the
demand for efficiently organizing these noisy scans has grown in parallel. To
this end, we propose a variational autoencoder (Kingma & Welling, 2014;
Rezende et al., 2014) for point clouds to identify continuous representations,
aligning with the continuous change and degradation of teeth over time.

Figure 1: VF-Net samples, generated by our probabilistic variational
autoencoder for point clouds.

Our solution is a probabilistic latent variable model that maintains a one-to-
one point correspondence between points in the observed and generated point
clouds. This one-to-one connection throughout the network allows for the
optimization of the original variational autoencoder objective. We project the
point cloud down to an intrinsic 2D surface representation to allow for
efficient sampling and mitigate the storage of information regarding the
overall shape in this space. These 2D projections impart a strong inductive
bias, proving highly beneficial when the input point cloud and the 2D surface
share topology. Simultaneously, creating a bottleneck prevents the model from
learning the identity mapping. Specifically, our _Variational Foldingnet (VF-
Net)_ learns a projection of the 3D point cloud input down to 2D space, which
then is deformed back into the input point cloud. Finally, these projections
facilitate mesh generation without further training, as well as
straightforward shape completion and shape extrapolation, all without
compromising the quality of the learned representations (see Fig. 1 for
samples).

Previous point cloud models generally lack one-to-one correspondence
throughout the network due to their invariant architecture design. Instead,
they evaluate reconstruction error using _Chamfer distances_ ($\mathrm{CD}$)
(Barrow et al., 1977) defined as

$\displaystyle\begin{split}D(\mathbf{A}\rightarrow\mathbf{B})&=\frac{1}{|\mathbf{A}|}\sum_{\mathbf{a}\in\textbf{A}}\min_{\mathbf{b}\in\mathbf{B}}\|\mathbf{a}-\mathbf{b}\|_{2},\\\
\textsc{Chamf-
Dist}(\mathbf{C},\mathbf{D})&=D(\mathbf{C}\rightarrow\mathbf{D})+D(\mathbf{D}\rightarrow\mathbf{C}).\\\
\end{split}$ (1)

Figure 2: VF-Net is a variational autoencoder with a normalizing flow prior
over the shape latent. Individual points are projected to 2D space,
establishing a one-to-one connection and facilitating mesh generation and
shape completion.

This metric solves the invariance problem. However, it also poses a new one:
_The Chamfer distance does not readily lead to a likelihood, preventing its
use in probabilistic modeling._ For instance, when used in the Gaussian
distribution, the function
$\mathbf{X}\mapsto\nicefrac{{1}}{{\mathcal{C}}}\exp(-\textsc{Chamf-
Dist}^{2}(\mathbf{X},\mu))$ cannot be normalized to have unit integral due to
the explicit minimization in Eq. 1. Consequently, previous latent variable
models are closer to regularized autoencoders than the variational
autoencoder. Since our model ensures one-to-one correspondence between points
in the point clouds, we can easily build a proper probabilistic model.

Moreover, to encourage further research, we release a new dataset, the _FDI 16
Tooth Dataset_ , which provides a large collection of dental scans, available
as both meshes and point clouds222Available here.. This dataset provides real-
world representations with simple topologies. We consider this an excellent
compromise between high-quality computer-aided design (CAD) models and sparse
LiDAR scans (Chang et al., 2015, 2017; Caesar et al., 2020; Armeni et al.,
2016). In digital dentistry, significant challenges are found in diagnostics,
tooth (crown) generation, shape completion of obstructed areas of the teeth,
and sorting point clouds.

In summary, we present the first fully probabilistic variational autoencoder
for point clouds, VF-Net, characterized by a highly expressive decoder with
state-of-the-art generative capabilities. All while learning compressed
representations and being adaptable for shape completion tasks. Furthermore,
we release a dataset of 7,732 tooth meshes to facilitate further research on
real-world 3D data.

##  2 Related work

We focus on point cloud representations of 3D objects, but there are many
alternative methods of representation including voxel grids (Zheng et al.,
2021; Wu et al., 2018), multi-angle inference (Wen et al., 2019; Han et al.,
2019), and meshes (Alldieck et al., 2019; Wang et al., 2018; Groueix et al.,
2018). A major paradigm in neural networks for point clouds is to remain
permutation and cardinality invariant. In terms of encoder-decoder models,
this frequently leads to designs without a one-to-one correspondence between
inputs and outputs (Yang et al., 2018; Groueix et al., 2018). This becomes an
obstacle in adapting the variational autoencoder to point clouds. Accordingly,
other methods have become prominent, including GANs (Li et al., 2018, 2019),
diffusion models (Zhou et al., 2021; Zeng et al., 2022), and traditional
autoencoders (Achlioptas et al., 2018; Groueix et al., 2018; Pang et al.,
2021).

Existing Point Cloud Variational Autoencoders. One attempt to design a
variational autoencoder for point clouds is SetVAE (Kim et al., 2021), which
uses transformers to process sets of point clouds. Their primary novelty is
introducing a latent space with an enforced prior inside the transformer
block. These transformer blocks are then stacked to form a hierarchical
variational autoencoder (Sønderby et al., 2016), posing issues in evaluating
its representations. However, the SetVAE is not fully probabilistic as the
reconstruction loss is approximated via Chamfer distances, resulting in a
model closer to a regularized autoencoder than a variational autoencoder. LION
(Zeng et al., 2022) is a latent variable model that maintains a one-to-one
mapping throughout the network, allowing for probabilistic evaluation.
However, they only implicitly utilize this by optimizing an L1-loss. Similar
to our work, they also encode their points, but instead of bottlenecking this,
they map them to a higher dimensional space. This, unfortunately, leads to
information about the shape being stored here, preventing direct
sampling/modification in this space. Finally, similarly to SetVAE, evaluating
the quality of representations in LION, a hierarchical latent variable model,
poses challenges. FrePolad (Zhou et al., 2023) is another latent diffusion
model similar to LION. Their primary novelty is the introduction of the
frequency rectification module that better captures high-frequency signals in
point clouds. They train their model via a modified VAE loss to account for
frequency rectified distances.

Other Generative Models. One fully probabilistic work is PointFlow (Yang et
al., 2019). PointFlow uses a continuous normalizing flow (CNF) both as prior
and decoder, similar to previous works seen on images (Kingma et al., 2017;
Sadeghi et al., 2019). Intuitively, one CNF can be considered modeling the
distribution of shapes, while the other models the point distribution given
the shape. Similarly, VF-Net’s encoder maps a global latent space, and the
point encoding projections offer a latent mapping for each input point.
However, PointFlow’s two CNFs are trained separately, while in VF-Net, these
are trained simultaneously. PointFlow is unfortunately very slow (Kim et al.,
2021). On our full proprietary dataset, PointFlow would have required 200 GPU
days of training. Thus, we excluded it from our baselines. Diffusion models
such as diffusion probabilistic model (DPM) (Luo & Hu, 2021) and point-voxel
diffusion (PVD) (Zhou et al., 2021) present two diffusion models for the point
clouds, especially PVD generates excellent new samples. However, diffusion
models do not find compressed structured representations of the data as our
VF-Net does; see table. LABEL:tab:method_properties for a model property
overview.

Figure 3:  _Top:_ Samples from the FDI 16 dataset with varying facet
regularity and resolution. _Bottom:_ VF-Net reconstructions.

Digital Dentistry. In computational dentistry, extrapolating the tooth’s
obstructed sides is a well-known task. Qiu et al. (2013) presents an
impressive attempt to use classic computational geometry methods. They attempt
to reconstruct the missing parts of the distal and mesial sides of the tooth.
This leads to a very smooth extrapolation, which performs well. Several works
within dentistry take this a step further, e.g., attempting to extrapolate not
just the sides but also the roots of the teeth (Wei et al., 2015; Zhou et al.,
2018; Wu et al., 2016). Unfortunately, CBCT scans are expensive and rare;
thus, we do not have a large enough dataset for neural network training.

##  3 Variational Point Cloud Inference

Background: FoldingNet. To handle varying sizes and arbitrary order in point
clouds, a common strategy is to employ neural networks exhibiting invariance
to changes in cardinality and permutation, as proposed by Qi et al. (2017) in
PointNet. FoldingNet employs a very similar encoder, $e$, that operates
independently on each point of the point cloud to identify the latent code,
$\mathbf{z}$. Subsequently, the folding-based decoder,
$f:\mathcal{Z}\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}$, "folds" a chosen
constant base shape, $\mathcal{G}$, according to the latent code, in our case
two-dimensional planar patch $[-1,1]^{2}$ (Yang et al., 2018). Both the
encoder, $e$, and the decoder, $f$, are jointly trained to minimize the
reconstruction error approximated via Chamfer distances (1),

$\displaystyle\mathcal{E}$ $\displaystyle=\textsc{Chamf-
Dist}\left(\mathbf{x}-f(e(\mathbf{x}),\mathcal{G})\right).$ (2)

This ensures invariance to cardinality and permutation changes, although it
complicates variational inference extensions. A variational autoencoder yields
a distribution for each input (Kingma & Welling, 2014; Rezende et al., 2014).
However, FoldingNet and most current permutation-invariant neural networks do
not have a correspondent output point for each input point in the point cloud.

###  3.1 The Variational FoldingNet

Following the variational autoencoder architecture, Variational FoldingNet
introduces a prior to the latent space. Our aim is to instill greater
structure to identify more robust representations. This also facilitates
improved handling of missing and noisy data, which is important for real-world
datasets. Motivated by unsupervised probabilistic representation learning’s
benefits across many tasks, including _generative modeling_ (Kingma & Welling,
2014; Rezende et al., 2014; Dinh et al., 2017; Ho et al., 2020), _out-of-
distribution detection_ (Nalisnick et al., 2019; Havtorn et al., 2021),
_handling missing data_ (Mattei & Frellsen, 2019) etc, we introduce
Variational FoldingNet (VF-Net). For a complete architecture overview, consult
Fig. 2.

One key feature is the introduction of a novel projection down to the planer
space, $[-1,1]^{2}$,
$\mathrm{proj}_{\mathcal{G}}(\mathbf{x}):\mathbb{R}^{3}\rightarrow\mathcal{G}$,
where $\mathbf{g}\in\mathcal{G}$. Note that no prior is placed upon the
projections, which we will refer to as our point encodings. We establish a
one-to-one correspondence throughout the entire network by decoding these
point encodings instead of a constant planar patch. _However, it also follows
that the projected points, $\hat{\mathcal{G}}$, are no longer constant or
independent of input $\mathbf{x}$. The folding of $\hat{\mathcal{G}}$ from the
planar space, $f(\hat{\mathcal{G}})$, remains determined by the parameter
vector $\mathbf{z}$ predicted by the PointNet encoder, $e$_. This projection
is learned together with the rest of the model,

$\displaystyle S$
$\displaystyle=f(\mathbf{z},\hat{\mathcal{G}}(\mathbf{x},\mathbf{z}))$ (3)
$\displaystyle\hat{\mathcal{G}}(\mathbf{x},\mathbf{z})$
$\displaystyle=\operatorname*{arg\,min}_{\hat{\mathbf{g}}\in\hat{\mathcal{G}}}\|\mathbf{x}-f(\mathbf{z},\hat{\mathbf{g}})\|^{2}.$
(4)

$S$ is the surface spanned by $f(\mathbf{z},\mathcal{\hat{G}})$. This is a
permutation invariant and cardinality invariant autoencoder with a one-to-one
point correspondence from end to end. This enables optimization through
traditional variational autoencoder methods.

To facilitate variational extension, we define the likelihood of the observed
data as $p(\mathbf{x})=\int
p(\mathbf{x}|\mathbf{z})p(\mathbf{z})\mathrm{d}\mathbf{z}$, which gives a
training objective. Unfortunately, the integral is generally intractable, and
approximations are necessary. Following conventional variational inference
(Kingma & Welling, 2014; Rezende et al., 2014), a lower bound (Elbo) on
$p(\mathbf{x})$

$\mathcal{L}\left(\mathbf{x}\right)=\mathbb{E}_{q\left(\mathbf{z}|\mathbf{x}\right)}\left[\log
p\left(\mathbf{x}|\mathbf{z}\right)\right]-\mathrm{KL}\left(q\left(\mathbf{z}|\mathbf{x}\right)\|p(\mathbf{z})\right),$

(5)

Table 2: Across five seeds, VF-Net produces close to as large a variety of teeth as PVD and LION while generating samples much closer to real teeth. MMD has been multiplied by 100. Method | MMD($\downarrow$) | COV(%$\uparrow)$ | 1-NNA(%$\downarrow)$
---|---|---|---
CD | EMD | CD | EMD | CD | EMD
Train subsampled | 21.00$\pm$0.09 | 51.53$\pm$0.06 | 49.00$\pm$0.64 | 46.95$\pm$2.79 | 49.83$\pm$0.68 | 50.97$\pm$0.82
SetVAE | 39.00$\pm$0.78 | 66.66$\pm$0.38 | 10.66$\pm$0.66 | 9.52$\pm$0.27 | 97.99$\pm$0.32 | 97.95$\pm$0.34
DPM | 20.71 $\pm$0.10 | 51.94$\pm$0.09 | 36.94$\pm$0.65 | 33.28$\pm$0.65 | 70.30$\pm$0.82 | 75.75$\pm$0.99
PVD | 21.58$\pm$0.03 | 51.64$\pm$0.08 | 44.11$\pm$0.76 | 43.23$\pm$0.92 | 62.85$\pm$0.78 | 60.70$\pm$1.06
LION | 22.12$\pm$0.15 | 52.75$\pm$0.12 |  45.12$\pm$0.60 |  43.32$\pm$1.28 | 68.56$\pm$0.73 | 66.76$\pm$0.94
VF-Net (Ours) |  20.38$\pm$0.09 |  49.72$\pm$0.04 | 42.85$\pm$0.64 | 40.20$\pm$0.71 |  56.31$\pm$0.39 |  56.05$\pm$0.32

where $q(\mathbf{z}|\mathbf{x})$ is an approximation to the posterior
$p(\mathbf{z}|\mathbf{x})$. Most current point cloud models replace the
likelihood with a distance metric here, making the models closer to a
regularized autoencoder. Our novel method for probabilistic evaluation for 3D
reconstruction networks avoids the computationally expensive Chamfer distance,
Eq. 1. In supplementary Fig. S1, we demonstrate that our projections can
effectively replace Chamfer distances. The two metrics closely align, with
Euclidean distances acting as an upper bound that tightens with improved
reconstruction precision. Other permutation invariant networks lack this one-
to-one connection between input and output, making such evaluation impossible.
Instead, they rely on Chamfer distances (1) (Yang et al., 2018; Groueix et
al., 2018; Kim et al., 2021), but no suitable normalization constant can be
derived for probabilistic distributions using Chamfer distances. Consequently,
probabilistic evaluation using Chamfer distances is impossible.

During the evaluation of the Elbo loss, we use a multivariate student-t
distribution with isotropic variance and three degrees of freedom as the
reconstruction term. This choice helps to decrease emphasis on outliers and
instead focus more on the majority of the data points.
$p(x|\mathbf{z},\mathbf{g})=\text{Student-t}(x|f(\mathbf{z},\mathbf{g}),\sigma^{2}(\mathbf{z},\mathbf{g})\mathbf{I},\nu)$,
where $f:\mathcal{Z}\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}$ and
$\sigma^{2}:\mathcal{Z}\times\mathbb{R}^{2}\rightarrow\mathbb{R}_{+}$ are
neural networks. No major changes were made to the generative process. We let
$p(\mathbf{z})$ be a normalizing flow prior over the parameters describing the
_shape_ of an object (Kingma et al., 2017). When the input, $\mathbf{X}$, and
the projections, $\mathcal{G}$, share topology, the bias allows for uniform
sampling in the planar patch $[-1,1]^{2}$. As in FoldingNet, this grid is
subsequently deformed according to $\mathbf{z}$. New samples can thus be
generated by first sampling $\mathbf{z}$ and then mapping the uniformly
sampled grid points through $f$ and $\sigma$,

$\displaystyle\mathbf{x}=f(\mathbf{z},\mathbf{g})+\sigma(\mathbf{z},\mathbf{g})\cdot\mathbf{t},\quad\mathbf{t}\sim\text{Student-t}(\mathbf{\nu}).$
(6)

This also enables straightforward mesh generation as deformations are smooth -
points projected closely to each other correspond to points close in output
space. Consequently, we can generate meshes by simply defining the facets in
the 2D planar space.

##  4 The FDI 16 Tooth Dataset

To improve the state-of-the-art modeling of dental scans, we will release an
extensive new dataset alongside this paper. The FDI 16 dataset is a collection
of 7,732 irregular triangle meshes of the right-side first maxillary molar
tooth formally denoted as ’ _FDI 16_ ’ following ISO 3950 notation (see Fig.
3). These meshes were acquired from fully anonymized intraoral scans primarily
scanned using 3Shape’s TRIOS 3 scanners. Each tooth in the FDI 16 Tooth
dataset was algorithmically segmented from an upper jaw scan by 3Shape’s Ortho
Systems 2023. As the teeth are a subsection of a full intraoral jaw scan,
there will be areas obstructed by the adjacent teeth. The teeth, therefore,
constitute open meshes and have clear boundaries with no representation of
interior object volume. All tooth meshes are from patients undergoing aligner
treatment, and accordingly, aligner attachments will be present in a
substantial number of scans. This introduces a bias towards younger
individuals, who generally have fewer restorations and dental problems. The
top row of Fig. 3 shows examples of such meshes. All scans have been made
publicly available fully anonymously as meshes and point clouds at millimeter
scale. The teeth have been algorithmically rotated to ensure that the $x$-axis
is turned towards the neighboring tooth (FDI 17) while the $y$-axis points in
the occlusal direction (direction of the biting surface). Finally, the
$z$-axis is given by the cross-product to ensure a right-hand coordinate
system.

Dental scans have a diverse set of research applications. This study explores
reconstruction, generation of new teeth, representation learning, and shape
completion. All of which have different but critical applications in digital
dentistry. We believe that the FDI 16 dataset addresses a crucial niche within
3D datasets by offering a dataset that strikes a balance between the highly
detailed but idealized CAD scans (Chang et al., 2015) and sparser real-world
LIDAR scans (Chang et al., 2017; Caesar et al., 2020; Armeni et al., 2016).
This dataset provides a valuable middle ground. Note that any method
considered for deployment must be capable of running efficiently on edge
devices without a significant performance overhead. This is particularly
important as intraoral scanners must function seamlessly even in areas with
limited network connectivity.

##  5 Experimental results

We next evaluate VF-Net’s performance on point cloud generation, auto-
encoding, shape completion, and unsupervised representation learning. Note
that FrePolad (Zhou et al., 2023) has been excluded from comparison as no
public implementation is available.

Figure 4: FoldingNet’s mesh reconstructions have gaps and highly distorted
facets. Conversely, VF-Net’s mesh facets are even more regular than the input
point cloud, and points in the reconstruction are placed closely resembling
its input.

Point cloud generation. To compare sampling performances, we deploy three
established metrics for 3D generative model evaluation (Yang et al., 2019).
Namely, minimum matching distance (MMD) is a metric that measures the average
distance to its nearest neighbor point cloud. Coverage (COV) measures the
fraction of point clouds in the ground truth test set that is considered the
nearest test sample neighbor for a generated sample. 1-nearest neighbor
accuracy (1-NNA) uses a 1-NN classifier to classify whether a sample is
generated or from the ground truth dataset, 50%, meaning generated samples are
indistinguishable from the test set. Data handling and training details for
FDI 16 experiments can be found in supplementary section S1.3 and S1.4,
respectively.

Table 3: Chamfer distances (CD) and earth mover’s distances (EMD) are multiplied by 100.  Method | FDI 16 Tooth | All FDIs
---|---|---
$\mathbf{CD}$ | $\mathbf{EMD}$ | $\mathbf{CD}$ | $\mathbf{EMD}$
DPM | 10.04 | 43.98 | 5.67 | 35.8
SetVAE | 21.50 | 59.24 | 9.98 | 51.48
LION | 5.35 | 22.85 | 3.02 | 9.66
FoldingNet | 5.26 | 33.67 | 3.43 | 31.25
VF-Net (ours) | 1.21 | 6.30 | 0.97 | 5.30

Sampling from VF-Net can be done by sampling a uniform grid in the latent
point encodings space, akin to FoldingNet. However, the corners of the uniform
grid cause edge artifacts in the generated samples, evident in generated
meshes in Fig. S2. The sampling metrics heavily punish such artifacts.
Instead, we trained a minor network similar to the decoder of FoldingNet to
predict the point encodings from the latent representation. We emphasize that
this is entirely unnecessary for regular sampling. The sampling evaluations
across five different seeds can be found in Table 2. The results demonstrate
that VF-Net generates much more accurate samples, as evidenced by the
significantly lower MMD and 1-NNA scores while being close in diversity to PVD
and LION (Zhou et al., 2021; Zeng et al., 2022). Furthermore, sampling is much
faster than PVD and LION as VF-Net does not depend on an iterative diffusion
process. Note that while MMD is very stable across seeds, the COV and 1-NNA
scores may vary quite a bit.

Outside of the FDI 16 dataset, we also train VF-Net on a proprietary dataset,
which includes the remaining teeth from the FDI 16 jaws; see supplementary
section S1.5 for training details. However, we did not quantify sampling
performance, as sampling evaluation on 40k test samples would be exceedingly
computationally expensive. We observe that VF-Net can sample from all major
teeth types, incisors, canines, premolars, and molars, see Fig. 1. Additional
mesh samples may be found in supplementary Fig. S2.

Point cloud auto-encoding. We evaluate VF-Net’s reconstruction quality to the
previously mentioned generative models and FoldingNet. This evaluation was
performed on both on FDI 16 dataset and the larger proprietary dataset. Please
consult supplementary sections S1.3 and S1.5 for data handling and training
details. We evaluated reconstructions using Chamfer distance and earth mover’s
distance (Rubner et al., 2000),

$\vspace{-2pt}\operatorname{EMD}(\mathbf{X},\mathbf{Y})=\min_{\phi:\mathbf{X}\rightarrow\mathbf{Y}}\sum_{\mathbf{x}\in\mathbf{X}}\|\mathbf{x}-\phi(\mathbf{x})\|_{2}.\vspace{-2pt}$
(7)

The earth mover’s distance measures the least expensive one-to-one
transportation between two distributions. However, this is computationally
expensive and thus rarely used for model optimization (Wu et al., 2021). The
reconstruction errors are presented in Table 3. Point-Voxel Diffusion (PVD)
(Zhou et al., 2021) was excluded from comparison due to not returning the same
tooth upon reconstruction.

Figure 5: Intra-point cloud relative predicted variance (red is high, green is
low). Notably, the carabelli cusp and aligner attachment areas exhibit high
variance, two features only present in a subset of individuals.

VF-Net achieves a significantly lower reconstruction error than our comparison
methods on both the FDI 16 dataset and the proprietary dataset comprising
119,496 teeth, encompassing 32 distinct teeth. As shown in Fig. 4, VF-Net’s
one-to-one correspondence is evident in its reconstruction. The point
placements mimic those in the input point cloud, while FoldingNet’s points are
evenly distributed. VF-Net and FoldingNet can both generate meshes without any
additional training of the model. In Fig. 4, deformation artifacts from
FoldingNet that were not as visible from its point cloud are evident.
FoldingNet folds the edge across the tooth to accommodate teeth of different
sizes. Besides mesh gaps, this also leads to highly irregular facets that
intersect one another. On the other hand, VF-Net can adjust the point encoding
area to avoid such artifacts. However, VF-Net’s reconstructions often exhibit
excessive smoothness and lack the desired level of detail. A common
observation in variational autoencoders (Kingma & Welling, 2014; Vahdat &
Kautz, 2021; Tolstikhin et al., 2019).

Figure 7: Interpolating between two teeth by interpolating their latent codes
using the same mesh decoding. Figure 6: _Left_ : Red points are removed from
the point cloud. _Right_ : Reconstruction and projected point encodings remain
highly similar despite point deletion. Sampling the missing area is
facilitated by sampling within the corresponding empty region of the latent
point encoding.

Variance estimation for point clouds. Predicted variances from the variance
network are shown in Fig. 5, where red indicates a higher variance and green
indicates a lower variance within each point cloud. Interestingly, the network
assigns higher variance to the fifth cusp and aligner attachments, features
only present in a subset of samples. Furthermore, the border of the mesh tends
to be assigned higher variance, likely due to a combination of data loading
and segmentation artifacts. When the network is not in doubt about the
previously mentioned two factors, the network assigns the highest variance to
the occlusal surface. All of which aligns with expectations of areas of the
teeth that have the most variance.

Simulated shape completion. One significant benefit of the inductive bias from
the point encodings is straightforward shape completion and shape
extrapolation. In computational dentistry, inferring the obstructed sides of a
tooth and reconstructing the tooth surface beneath obstructions such as braces
pose a key challenge. Paired data of obstructed and unobstructed surfaces is
exceedingly rare. Therefore, developing a model capable of extrapolating such
surfaces without explicit training is highly desirable. To this end, we
simulate the task by evaluating the interpolation performance of each model.
This is done by sampling a point on the outward side of the tooth and deleting
its nearest neighbors to a total of 200 points. Selecting a mid-buccal point
simulates bracket removal prediction ("Bracket sim") while opting for a lower
buccal point simulates the obstructed side prediction ("Gap sim").

Figure 8: Moving toward tooth wear in the latent space. _Left_ : Highlighted
in red are areas with higher values than the original reconstruction. _Middle_
: The original reconstruction. _Right_ : Areas depicted in blue are lower than
the original mesh. In general, as the level of tooth wear increases, we
observe a gradual smoothing in the occlusal surface.

An example of a synthetic hole is depicted in Fig. 6, where the red points are
to be removed. Both reconstructions and latent point encodings remain highly
similar despite the removal of the red points. Extrapolation/interpolation can
be performed by sampling in the point encoding space. To quantify the
interpolation performance, we calculate the distance from the deleted points
to their nearest neighbor in the completed point cloud; see supplementary Sec.
S1.7 for more experiment details. To contextualize the performance, we trained
several shape completion methods (PVD (Zhou et al., 2021), PoinTr (Yu et al.,
2021), VRCNet (Pan et al., 2021)). Since these methods only predict the
missing area, a completely fair comparison cannot be made. The results can be
found in Table 4, under "Bracket sim" and "Gap sim," simulating the removed
bracket and the gap between teeth, respectively. Here, VF-Net outperforms its
peers when it comes to untrained interpolation, and as expected there is a gap
in performance between the trained and untrained methods. Shape completion
using LION’s latent points from the original tooth contains information about
the shape, rendering a fair comparison infeasible.

Table 4: Unsupervised generative models in the top half are untrained interpolation, while the bottom half are trained models. All Chamfer distances have been multiplied by 100.  | Method | Bracket sim | Gap sim
---|---|---|---
Unsupervised | DPM | 15.88 | 38.00
SetVAE | 11.50 | 13.35
FoldingNet | 16.42 | 20.14
VF-Net (ours) | 4.35 | 3.55
Supervised | PVD | 2.23 | 2.37
PoinTr | 1.84 | 1.83
VRCNet | 2.42 | 2.04

Representation learning. We compare our latent representation to FoldingNet’s,
as it is the comparison model with the most interpretable latent variables.
First, we follow FoldingNet’s proposed evaluation method of classifying the
input point cloud from the latent space. Using a linear support vector machine
(SVM) to classify which tooth from the larger proprietary dataset is embedded,
a 32-class problem. Here, the SVM achieves 96.80% accuracy on VF-Net’s latent
codes compared to 96.36% of FoldingNet. Indicating all global point cloud
information is stored in the latent variables, meaning the latent point
encodings exclusively contain information about specific points. No
information pertaining to the overall point cloud shape is stored in the point
encodings. For qualitative assessment, an interpolation between two FDI 16
teeth and an interpolation example between an incisor and a premolar can be
found in Fig. 7. Both interpolations exhibit a seamless transition in the
latent space; for a more detailed view, see supplementary Fig. S3.

Next, we attempt to add and remove toothwear; see Fig. 8. We navigate the
latent space of VF-Net in the direction of tooth wear or away from it. The
direction was determined by calculating the average change in latent
representations when encoding 10 teeth from their counterparts with
synthetically induced tooth wear. These teeth were manually sculpted to
simulate tooth wear; see supplementary Fig. S4. We observe behavior that
closely aligns with our expectations of how the tooth would change when adding
or subtracting tooth wear.

To quantify the performance, we train a small PointNet model (Qi et al., 2017)
on a proprietary dataset of 1400 teeth annotated with light/medium/heavy tooth
wear. Subsequently, validate whether a change in the latent space yielded the
expected change in classifier prediction. In Table 5, each class denotes the
base class before adding/removing tooth wear. For light and heavy, we added
and removed tooth wear, respectively, while medium tooth wear teeth were
evaluated both when adding/removing wear. The findings presented in Table 5
indicate that VF-Net’s latent representations show greater robustness.

Table 5: The percentage of teeth with the expected change in classification prediction when changing the latent code towards/away from the tooth wear direction. L, M, and H denote light, medium, and heavy wear. Method | L | M | H
---|---|---|---
FoldingNet | 94.95% | 91.77% | 97.8%
VF-Net (ours) | 97.98% | 96.31% | 98.68%

Limitations. Until now, the inductive bias from folding a 2D plane to a point
cloud has proven highly beneficial. This is only the case when the input point
cloud shares topology with the 2D plane. Unfortunately, this inductive bias is
not as beneficial when the two topologies differ. We trained VF-Net on
ShapeNet data (Chang et al., 2015). The drawback is not evident through the
reconstructions; see supplementary Table S1. VF-Net has a low reconstruction
error, but LION boasts the lowest. Issues arise when attempting to generate
new samples. Due to information of the shape being stored in the latent point
encodings, as depicted in Fig. 9. The latent point encodings form a non-
continuous distribution, posing challenges for sampling new models. Note that
for point clouds sharing topology, VF-Net is strongly biased towards
generating a continuous distribution; see Fig. 9. Addressing this issue could
involve training a flow or diffusion prior for the point encodings, similar to
the approach used in LION (Zeng et al., 2022). However, since this was not the
focus of our model, we did not pursue this idea.

Figure 9: _Left_ : While accurately reconstructed, the airplane forms a non-
continuous distribution in the latent point encoding, posing challenges for
sampling. _Right:_ An incisor and its corresponding point encodings.

##  6 Conclusion

We have introduced the _FDI 16_ dataset and _Variational FoldingNet (VF-Net)_
, a fully probabilistic point cloud model in the same spirit as the original
variational autoencoder (Kingma & Welling, 2014; Rezende et al., 2014). The
key technical innovation is the introduction of a point-wise encoder network
that replaces the commonly used Chamfer distance, allowing for probabilistic
modeling. Importantly, we have shown that VF-Net offers better auto-encoding
than current state-of-the-art generative models and more realistic sample
generation for dental point clouds. Additionally, VF-Net offers
straightforward shape completion and extrapolation due to its latent point
encodings. All while identifying highly interpretable latent representations.

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##  Appendix S1 Supplementary Material

###  S1.1 Chamfer vs Euclidean

Figure S1: In the depicted plot, it is evident that the Euclidean distance
serves as an upper limit for the Chamfer distance. In most instances,
optimization with VF-Net results in nearly identical distances. This empirical
finding supports our claim that the Chamfer distance can be efficiently
replaced with an appropriate encoder choice.

###  S1.2 ShapeNet Reconstruction Performances

Table S1: Both Chamfer distances (CD) and earth mover’s distances (EMD) are multiplied by 1000. Lower values indicate better reconstruction performance. Method | ShapeNet Airplanes
---|---
$\mathbf{CD}$ | $\mathbf{EMD}$
DPM | 0.18 | 47.82
SetVAE | 0.14 | 30.60
PVD | 0.31 | 90.45
LION | 0.025 | 7.30
FoldingNet | 0.079 | 31.47
VF-Net (ours) | 0.039 | 7.90

###  S1.3 Data Handling

For dental scan experiments, the point clouds used were constructed from
vertices and facet midpoints. The cardinality of the raw point clouds varied
significantly, ranging from around 2,000 to 65,000 points. To handle this, we
subsampled 2048 points from each point cloud during training. As FoldingNet
and VF-Net deform from a set space, selecting an appropriate normalization
method is crucial. The normalization determines the amount of deformation
needed for initial points to reach the desired final reconstruction. As we
have defined our planar patch as $[-1,1]^{2}$, we scale the data so 99.5%

###  S1.4 FDI 16 Training Details

VF-Net was trained using an Adamax optimizer with an initial learning rate of
0.001. Each backpropagation iteration utilized a batch size of 64, and the
training persisted for 16,000 epochs. We employed a KLD warm-up (Sønderby et
al., 2016) during the initial 4,000 epochs, whenever applicable. During the
initial training of VF-Net, a constant variance was used. Following that, a
distinct training phase of 100 epochs was explicitly conducted to train the
variance network.(Detlefsen et al., 2019).

###  S1.5 All FDI Training Details

In the experiments conducted on the proprietary dataset, encompassing all
teeth, each model maintained the same architecture and size as employed in the
FDI 16 experiment. However, training was confined to 1,250 epochs,
incorporating a KLD warm-up phase constituting one-fourth of the total epochs
when relevant. Again, a separate training run of 100 epochs was done to tune
the variance network (Detlefsen et al., 2019).

###  S1.6 Sampled All FDI Teeth

Figure S2: A display of meshes sampled by VF-Net showcases comprehensive
coverage across the four major tooth modalities: Incisor, canine, pre-molar,
and molar. These represent the primary types of teeth in the proprietary
dataset.

###  S1.7 Shape Completion Experiments

In the shape completion experiments, each model is permitted to sample three
times the standard number of points. Unsupervised models, therefore, have the
allowance to sample 6,144 points each, as their sampling is not limited to the
missing area. Conversely, for supervised models trained to predict the 200
missing points, we extract 600 points in this region to balance point density
between supervised and unsupervised models. The evaluation is done by
calculating a one-directional Chamfer distance (1) from predicted points to
ground truth, quantifying shape completion performance.

###  S1.8 Interpolation

Figure S3: Interpolating between two teeth by interpolating their latent codes
using the same mesh decoding.

###  S1.9 Synthetic Toothwear Teeth

Figure S4: Two of the ten manually sculpted teeth to simulate tooth wear.
_Left_ : Areas with higher values compared to the original mesh are
highlighted in red. _Middle_ : The original mesh. _Right_ : The mesh showcases
areas depicted in blue that are lower than the original mesh.