# Centroids and equilibrium points of convex bodies

Zsolt Lángi  and  Péter L. Várkonyi  Zsolt Lángi, Department of Algebra and
Geometry, Budapest University of Technology and Economics,
Műegyetem rkp. 3., H-1111 Budapest, Hungary, and
Alfréd Rényi Institute of Mathematics,
Reáltanoda utca 13-15, H-1053, Budapest, Hungary<EMAIL_ADDRESS>Péter L.
Várkonyi, Department of Mechanics, Materials, and Structures, Budapest
University of Technology and Economics,
Műegyetem rkp. 3., H-1111 Budapest, Hungary<EMAIL_ADDRESS>
###### Abstract.

The aim of this note is to survey the results in some geometric problems
related to the centroids and the static equilibrium points of convex bodies.
In particular, we collect results related to Grünbaum’s inequality and the
Busemann-Petty centroid inequality, describe classifications of convex bodies
based on equilibrium points, and investigate the location and structure of
equilibrium points, their number with respect to a general reference point as
well as the static equilibrium properties of convex polyhedra.

###### Key words and phrases:

centroid, equilibrium point, Grünbaum’s inequality, Busemann-Petty centroid
inequality, Gömböc, monostable polyhedra

###### 2020 Mathematics Subject Classification:

52A20, 52A38

The authors thank Timea Szabó and Gábor Domokos for their permissions to use
their images of bodies in various equilibrium classes (Fig. 1). Zsolt Lángi is
partially supported by the ERC Advanced Grant “ERMiD”, and the NKFIH grant
K147544. P. L. Várkonyi is supported by the NKFIH grant 143175.

##  1\. Introduction

The investigation of the centroids and static equilibrium points of solids has
been in the focus of research since ancient Greece. These concepts play an
important role in many different disciplines from physics to engineering to
geology, and also appear in many problems in pure mathematics. The aim of this
paper is to collect the results in the geometric literature related to some of
these problems.

Apart from a few remarks, the objects of our paper are convex bodies in the
$d$-dimensional Euclidean space $\mathbb{R}^{d}$, defined as compact, convex
sets with nonempty interior. We denote the standard inner product of this
space by $\langle\cdot,\cdot\rangle$, and the Euclidean norm of a vector
$p\in\mathbb{R}^{d}$ by $|p|=\sqrt{\langle p,p\rangle}$. The _closed segment_
with endpoints $p,q\in\mathbb{R}^{d}$ is denoted as $[p,q]$. An important
convex body in $\mathbb{R}^{d}$ is the closed unit ball centered at the origin
$o$, which we denote by $\mathbf{B}^{d}$. The boundary of this ball is denoted
by $\mathbb{S}^{d-1}$. We use the standard notation $\operatorname{int}(X)$,
$\operatorname{bd}(X)$, $\operatorname{conv}(X)$, $\operatorname{vol}_{d}(X)$
for the _interior, boundary, convex hull_ and $d$-dimensional _volume_ of a
set $X\in\mathbb{R}^{d}$. Two-dimensional volume is also called _area_ ,
denoted by $\operatorname{area}(\cdot)$, and $3$-dimensional volume is denoted
by $\operatorname{vol}(\cdot)$. For any $s>0$, we set
$\kappa_{s}=\frac{\pi^{s/2}}{\Gamma\left(\frac{s}{2}+1\right)}$, where
$\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\,dt$ is the usual Gamma function.
Then, for any nonnegative integer $d$,
$\kappa_{d}=\operatorname{vol}_{d}(\mathbf{B}^{d})$.

As usual, the orthogonal complement of a linear subspace $L$ of
$\mathbb{R}^{d}$ is denoted by $L^{\perp}$, and the orthogonal projection of a
set $X$ onto $L$ is denoted by $X|L$. For brevity, if $L$ is a line generated
by a nonzero vector $u$, we set $u^{\perp}=L^{\perp}$, $K|u=K|L$, and
$K|u^{\perp}=K|L^{\perp}$. Finally, for any $u\in\mathbb{S}^{d-1}$ and
$\alpha\in\mathbb{R}$, we denote the hyperplane $\\{x\in\mathbb{R}^{d}:\langle
x,u\rangle=\alpha\\}$ by $H(u,\alpha)$. In this notation $u$ is a _normal
vector_ of $H(u,\alpha)$ and $|\alpha|$ is its distance from $o$.

Let $K$ be a convex body with $o\in\operatorname{int}(K)$. The _radial
function_ $\rho_{K}:\mathbb{S}^{d-1}\to\mathbb{R}$ of $K$ is defined as

(1.1) $\rho_{K}(u)=\max\\{\alpha\in\mathbb{R}:\alpha u\in K\\},$

and its _support function_ $h_{K}:\mathbb{S}^{d-1}\to\mathbb{R}$ as

(1.2) $h_{K}(u)=\max\\{\langle x,u\rangle:x\in K\\}.$

The latter function is often defined for any $u\in\mathbb{R}^{d}$ using the
formula in (1.2). To distinguish these two functions, we use the notation
$\bar{h}_{K}:\mathbb{R}^{d}\to\mathbb{R}$ for the one extended to
$\mathbb{R}^{d}$. We note that according to our definition, the function
$\bar{h}_{K}$ is a positively homogeneous, convex, nonnegative function which
is zero only at $o$. It is also known [100] that any function satisfying these
properties is the support function of some convex body $K$ with
$o\in\operatorname{int}(K)$.

A set $S\subseteq\mathbb{R}^{d}$ is called _star-shaped_ with respect to $p$,
if for any $q\in S$, we have $[p,q]\subseteq S$. In particular, a convex set
$K$ is star-shaped with respect to any $p\in K$. We define the radial function
$\rho_{S}:\mathbb{S}^{d-1}\to\mathbb{R}$ of any compact set, star-shaped with
respect to $o$, by the formula in (1.1). Such a set is called a _star body_ if
its radial function is strictly positive and continuous.

In the remaining part of the paper we present the results related to our basic
concepts. In particular, in Section 2 we investigate the centroids of convex
bodies, and discuss results related to two important problems concerning them:
Grünbaum’s centroid inequality in Subsection 2.1, and the Busemann-Petty
centroid inequality in Subsection 2.2. In Section 3 we study various problems
related to the static equilibrium points of convex bodies. Within this, in
Subsection 3.1 we present a classification system of convex bodies based on
the number of their equilibria. In Subsection 3.2 we refine this system by
distinguishing convex bodies based on the topological properties of the
gradient flow of their radial functions. In Subsection 3.3 we investigate how
difficult it is to move a convex body from one class to another one. In
Subsections 3.4, 3.5 and 3.6 we deal with the locations of equilibrium points
on a convex body, equilibria with respect to a general reference point, and
equilibria of convey polyhedra. Finally, in Section 4, we collect some
applications and problems related to the presented topics.

##  2\. The centroid of a convex body

The main concept of this section is the following.

######  Definition 2.1.

Let $X$ be a compact set in $\mathbb{R}^{d}$ with nonempty interior. Then the
_centroid_ of $K$ is the point

$c(X)=\frac{1}{\operatorname{vol}_{d}(X)}\int_{X}x\,dx.$

In the above definition, if $c(X)=o$, we say that $X$ is _centered_. A concept
related to the centroid of a set is its first moment with respect to a
hyperplane.

######  Definition 2.2.

Let $X$ be a compact set in $\mathbb{R}^{d}$ with nonempty interior, and let
$u\in\mathbb{S}^{d-1}$, $\alpha\in\mathbb{R}$. Then the quantity

(2.1) $\int_{X}\left(\langle u,x\rangle-\alpha\right)\,dx$

is called the _first moment_ of $K$ with respect to the hyperplane
$H(u,\alpha)$.

We note that the quantity $\langle u,x\rangle-\alpha$ in (2.1) is equal to the
signed distance of $x$ from $H(u,\alpha)$, according to the orientation
induced by $u$. From Definitions 2.1 and 2.2 it follows that the first moment
of $X$ with respect to $H(u,\alpha)$ is equal to
$\operatorname{vol}_{d}(X)c(X)$, that is, it is equal to that of a point mass
at $c(X)$ with weight $\operatorname{vol}_{d}(X)$. While the notion of
centroid in non-Euclidean spaces is outside the scope of this paper, we remark
that an axiomatic generalization of centroid in various manifolds including
hyperbolic and spherical spaces, based partly on this observation, can be
found in [46]. For more information on centroids in spherical or hyperbolic
spaces, the interested reader is referred to [40, 43, 45, 8].

The geometric concept of centroid is partly motivated by the mechanics of
rigid bodies. In this context the centroid of a convex body $K$ coincides with
the _center of mass_ of the solid $K$ in the case of homogeneous mass
distribution, as well as with the _center of gravity_ of $K$ in a uniform
gravitational field. However a physical object may also have inhomogeneous
mass distribution, in which case the center of mass and the center of gravity
of $K$ are identical to each other but different from the centroid, as they
can be any interior point of $K$ depending on the mass distribution.

###  2.1. Grünbaum’s centroid inequality

Consider a convex disk $K\subset\mathbb{R}^{2}$, and let $L$ be a line through
its centroid. This line dissects $K$ into two convex disks $K_{1}$ and
$K_{2}$. It was observed by Winternitz in 1923 [12] that for any such disk $K$
and line $L$, we have

$\frac{4}{5}\leq\frac{\operatorname{area}(K_{1})}{\operatorname{area}(K_{2})}\leq\frac{5}{4},$

with equality if $K$ is a triangle.

This result was generalized by Grünbaum [53] for any dimension in 1960 as
follows.

######  Theorem 2.3.

Let $K\subset\mathbb{R}^{d}$ be a centered convex body. For any
$u\in\mathbb{S}^{d-1}$ let $K_{u}$ denote the intersection of $K$ with the
closed half space bounded by $u^{\perp}$ and containing $u$. Then, for any
$u\in\mathbb{S}^{d-1}$, we have

$\frac{\operatorname{vol}_{d}(K_{u})}{\operatorname{vol}_{d}(K_{-u})}\leq\left(1+\frac{1}{d}\right)^{d}-1.$

Based on Theorem 2.3, let
$\theta(K)=\sup\left\\{\frac{\operatorname{vol}_{d}(K_{u})}{\operatorname{vol}_{d}(K_{-u})}:u\in\mathbb{S}^{d-1}\right\\}$.
This quantity can be regarded as a measure of symmetry of $K$: it satisfies
$\theta(K)\geq 1$ for every convex body $K$, where we have equality for
centrally symmetric bodies, and it attains its maximal value
$\theta_{d}=\left(1+\frac{1}{d}\right)^{d}-1$ e.g. for convex cones.

In 2000, Groemer [52] proved a stability version of Grünbaum’s result. Namely,
he proved that for every $\varepsilon\geq 0$, there is some $\lambda_{d}>0$
depending only on $d$ such that if $\theta(K)\geq\theta_{d}-\varepsilon$ for
some convex body $K$ in $\mathbb{R}^{d}$, then there is a convex cone $C$ in
$\mathbb{R}^{d}$ such that

$\operatorname{vol}_{d}(K\Delta
C)\leq\lambda_{d}\operatorname{vol}_{d}(K)\varepsilon^{1/(2d^{2})},$

where $X\Delta Y$ denotes the symmetric difference of $X,Y$.

Using the inequality $\left(1+\frac{1}{d}\right)^{d}<e$, Theorem 2.3 yields a
simple dimension-free estimate for the volumes of the intersections: for any
convex body $K$ in $\mathbb{R}^{d}$, and for any closed half space $H$
containing the centroid of $K$ in its boundary, $\operatorname{vol}_{d}(H\cap
K)\geq\frac{1}{e}\operatorname{vol}_{d}(K)$. This inequality was generalized
by Bertsimas and Vempala [7] for a convex body $K$ in _isotropic position_ ,
defined as a special type of centered convex body: for any unit vector
$u\in\mathbb{S}^{d-1}$, $K$ satisfies the equality
$1=\frac{1}{\operatorname{vol}_{d}(K)}\int_{K}(\langle u,x\rangle)^{2}\,dx$.
Their result states that if $K$ is a convex body in $\mathbb{R}^{d}$ in
isotropic position, and $z\in K$ is at distance $t$ from the origin, then for
any closed half space $H$ containing $z$, $\operatorname{vol}_{d}(H\cap
K)\geq\left(\frac{1}{e}-t\right)\operatorname{vol}_{d}(K)$. Here we note that
for any centered convex body $K$ there is a linear transformation $A$ such
that $A(K)$ is in isotropic position (see e.g. [7]). Since linear
transformations do not change the ratio of volumes and the lengths of parallel
segments, it seems possible to restate the result in [7] for an arbitrary
convex body $K$ centered at $o$. This was done in [102] by Shyntar and Yaskin,
who strengthened of the result in [7] for $\mathbb{R}^{2}$ in the following
form: Let $K$ be a plane convex body with $o$ as its centroid. Let
$-1<\alpha<2$ and $u\in\mathbb{S}^{1}$. Let $H$ be the closed half plane
defined as $H=\\{x\in\mathbb{R}^{2}:\langle x,u\rangle\geq\alpha
h_{K}(-u)\\}$. Then

(2.2) $C_{1}(\alpha)\operatorname{area}(K)\leq\operatorname{area}(K\cap H)\leq
C_{2}(\alpha)\operatorname{area}(K),$

where

$C_{1}(\alpha)=\left\\{\begin{array}[]{l}\frac{1}{9}(2-\alpha)^{2},\hbox{ if
}\alpha\in(-1,0),\\\ \frac{4}{9}(1+\alpha)(1-2\alpha),\hbox{ if
}\alpha\in(0,1/2),\\\ 0,\hbox{ if }\alpha\in(1/2,2),\end{array}\right.$

and

$C_{2}(\alpha)=\left\\{\begin{array}[]{l}1-\frac{4}{9}(1+\alpha)^{2},\hbox{ if
}\alpha\in(-1,0),\\\ \frac{5-3\alpha}{9(1+\alpha)},\hbox{ if
}\alpha\in(0,1),\\\ \frac{1}{9}(2-\alpha)^{2},\hbox{ if
}\alpha\in(1,2).\end{array}\right.$

The authors of [102] also characterized the equality cases in (2.2).

Observe that if $K$ is a centered convex body in $\mathbb{R}^{d}$, and $H$ is
a hyperplane through $o$, then $o$ may not be the centroid of $K|H$. Thus, it
is a meaningful question to examine Grünbaum’s problem for projections of
convex bodies. This problem was completely solved by Stephen and Zhang [107],
who proved that for any centered convex body $K$ in $\mathbb{R}^{d}$, any
$k$-dimensional linear subspace $E$ with $1\leq k\leq d$, and any half space
$H$ bounded by $u^{\perp}$ with $u\in E\cap\mathbb{S}^{d-1}$, we have

$\operatorname{vol}_{k}((K|E)\cap
H)\geq\left(\frac{k}{d+1}\right)^{k}\operatorname{vol}_{k}(K|E).$

Furthermore, here there is equality for some $K$ and $E$ if and only if
$K=\operatorname{conv}\\{y_{1}+L_{1},y_{2}+L_{2}\\}$, where

  * •

$L_{1}\subset u^{\perp}$ and $L_{1}|(E\cap u^{\perp}$ are $(k-1)$-dimensional
convex bodies,

  * •

$L_{2}\subset E^{\perp}$ is a $(d-k)$-dimensional convex body,

  * •

$y_{1},y_{2}\in\mathbb{R}^{d}$ satisfy $y_{1}\notin H$ and
$y_{2}\in\operatorname{int}(H)$,

  * •

the centroid of $K$ is $o$.

More specifically, the authors of [107] prove a more general version of the
above result for volumes, where the centroid of $K$ is replaced by an
arbitrary point in the interior of $K$, representing the center of mass of an
inhomogeneous object.

In the same vein, one can examine Grünbaum’s problem for sections of a convex
body. This was proposed by Fradelizi, Meyer and Yaskin [44], who proved that
for any centered convex body $K$, linear subspace $V$ of dimension $d-k$ with
$0\leq k\leq d-1$, and closed half space $H$ with boundary $u^{\perp}$ for
some $u\in V\cap\mathbb{S}^{d-1}$, we have

(2.3) $\operatorname{vol}_{d-k}(K\cap V\cap H)\geq
c\operatorname{vol}_{d-k}(K\cap V),$

where $c\geq\frac{c_{0}}{(k+1)^{2}}\left(1+\frac{k+1}{d-k}\right)^{-(d-k-2)}$
for some absolute constant $c_{0}>0$. This result follows from a more general
setting investigated in [44] for intersections of a convex body $K$ with a
convex cone, which the authors use to prove a conjecture of Meyer and Reisner
[84] about the volumes of convex intersection bodies. Grünbaum’s problem for
sections was settled in [87] by Myroshnychenko, Stephen and Zhang, who proved
the following: for any convex body $K$ in $\mathbb{R}^{d}$, a $k$-dimensional
linear subspace $E$ of $\mathbb{R}^{d}$ and a closed half space $H$ bounded by
$u^{\perp}$ with $u\in\mathbb{S}^{d-1}\cap E$, if the centroid of $K$ lies in
$E\cap u^{\perp}$, then

$\operatorname{vol}_{k}(K\cap E\cap
H)\geq\left(\frac{k}{d+1}\right)^{k}\operatorname{vol}_{k}(K\cap E).$

Furthermore, here we have equality if and only if
$K=\operatorname{conv}\left\\{-\left(\frac{d-k+1}{k}\right)z+D_{0},z+D_{1}\right\\}$,
where

  * •

$z\in E\cap\operatorname{int}(H)$,

  * •

$D_{0}$ is a $(k-1)$-dimensional convex body in $E\cap u^{\perp}$,

  * •

$D_{1}$ is a centered $(d-k)$-dimensional convex body in a $(d-k)$-dimensional
subspace $F$ of $\mathbb{R}^{d}$ such that $E\cup F$ spans $\mathbb{R}^{d}$.

The results about both the intersections and the projections were generalized
for dual volumes by Stephen and Yaskin [106] as follows. Let $K$ be a centered
convex body in $\mathbb{R}^{d}$ with $o$, and let $E$ be a $k$-dimensional
linear subspace of $\mathbb{R}^{d}$, where $1\leq k\leq d$. Let $H$ be a
closed half space bounded by $u^{\perp}$ with $u\in\mathbb{S}^{d-1}\cap E$.
Then, for any $1\leq i\leq k$,

$\tilde{V}_{i}((K|E)\cap
H)\geq\left(\frac{i}{d+1}\right)^{i}\tilde{V}_{i}(K|E),$

and

$\tilde{V}_{i}(K\cap E\cap
H)\geq\left(\frac{i}{d+1}\right)^{i}\tilde{V}_{i}(K\cap E),$

where $\tilde{V}_{i}(\cdot)$ denotes $i$th dual volume with respect to the
$k$-dimensional subspace $E$. It is worth noting that the case $i=k$ of these
statements coincides with the results in [107] and [87], respectively,
mentioned in the previous two paragraphs.

An interesting discrete version of the theorem of Winternitz was proposed by
Shyntar and Yaskin [102]. To state the problem, for any set
$X{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\subseteq}\mathbb{R}^{2}$
we denote the cardinality of the point set $X\cap\mathbb{Z}^{2}$ by $\\#X$,
and we call a convex polygon $P$ an _integer polygon_ if all the vertices of
$P$ have integer coordinates. In this setting, the problem of Winternitz asks
for finding the largest positive number $C>0$ such that for any integer
polygon $P$, and any closed half plane $H$ containing $c(P)$ in its boundary,
we have $\frac{\\#(K\cap H)}{\\#K}\geq C$. The authors of [102] give an
example showing that there is no positive number $C$ that satisfies this
property, and prove that, for any integer polygon $P$ and any closed half
plane containing $c(3P)$ in its boundary,

$\frac{\\#((3P)\cap H)}{\\#(3P)}>\frac{1}{6}.$

Furthermore, for any $t\geq 6$, any integer polygon $P$ and any closed half
space containing $c(tP)$ in its boundary, we have

(2.4) $\frac{\\#((tP)\cap
H)}{\\#(tP)}\geq\frac{\frac{4}{9}t^{2}-2t-3}{t^{2}+3t+2}.$

Here we note that for $t>0$, the expression on the right-hand side of (2.4) is
increasing, and tends to $\frac{4}{9}$ as $t\to\infty$.

A generalization of Grünbam’s theorem for not necessarily convex sets was
considered by Marín Sola and Yepes Nicolás [82]. To state their result, we
recall that by Brunn’s theorem, for any compact, convex set $K$ in
$\mathbb{R}^{d}$, and any $u\in\mathbb{S}^{d-1}$, the function
$f_{X}:\mathbb{R}\to\mathbb{R}$,
$f(t)=\operatorname{vol}_{d-1}(K\cap(tu+u^{\perp}))$ is
$\frac{1}{d-1}$-concave on its support. The authors of [82] proved that if
$X\subset\mathbb{R}^{d}$ is a centered compact set with nonempty interior, and
$H$ is a closed half space with $u^{\perp}$ as its boundary for some
$u\in\mathbb{S}^{d-1}$, and the function $f_{X}:\mathbb{R}\to\mathbb{R}$,
$f(t)=\operatorname{vol}_{d-1}(X\cap(tu+u^{\perp}))$ is $p$-concave on its
support for some $p>0$, then

$\frac{\operatorname{vol}_{d}(X\cap
H)}{\operatorname{vol}_{d}(X)}\geq\left(\frac{p+1}{2p+1}\right)^{(p+1)/p},$

and if $f_{X}$ is log-concave on its support, then

$\frac{\operatorname{vol}_{d}(X\cap
H)}{\operatorname{vol}_{d}(X)}\geq\frac{1}{e}.$

They also characterize the equality case. The same problem, under the more
general condition that the function $f_{X}$ is $\varphi$-concave for some
strictly increasing, continuous function
$\varphi:[0,\infty)\to\mathbb{R}\cup\\{-\infty\\}$ satisfying some mild
regularity conditions, was investigated by Marín Sola [81].

###  2.2. The Busemann-Petty centroid inequality

Let $K$ be a centered convex body in $\mathbb{R}^{d}$. Consider the function
$H_{K}:\mathbb{R}^{d}\to\mathbb{R}$, defined by

(2.5) $H_{K}(u)=\frac{1}{\operatorname{vol}_{d}(K)}\int_{K}|\langle
x,u\rangle|\,dx.$

It is well known that this function is positively homogeneous, nonnegative,
convex function which is zero only at $o$ [95]. Thus, $H_{K}$ is the support
function of a convex body $CK$ (see Section 1), called the _centroid body_ of
$K$. This body is strictly convex, and if $K$ is symmetric to $o$, i.e.
$-K=K$, then the hypersurface $\operatorname{bd}(CK)$, called the _centroid
surface_ of $K$, coincides with the set of the centroids of the intersections
of $K$ with the closed half spaces containing $o$ in their boundaries.
Centroid bodies, attributed by Blaschke to Dupin (see e.g. [100]), were given
their name in the seminal work of Petty [95], who proved the inequality that

(2.6)
$\operatorname{vol}_{d}(CK)\geq\left(\frac{(d+1)\kappa_{d}}{2\kappa_{d-1}}\right)^{d}\operatorname{vol}_{d}(K),$

with equality if and only if $K$ is an $o$-symmetric ellipsoid. This
inequality is called the _Busemann-Petty centroid inequality_. In 2016, Ivaki
[63] proved a stability version of the equality case of this inequality in the
plane. Namely, he proved that there are positive constants $\varepsilon_{0}$
and $\gamma$ such that for any $0<\varepsilon<\varepsilon_{0}$ and any plane
convex body $K$ satisfying
$\frac{\operatorname{area}(CK)}{\operatorname{area}(K)}\leq\left(\frac{4}{3\pi}\right)^{2}(1+\varepsilon)$,
the Banach-Mazur distance of $K$ and an ellipsoid is at most
$1+\gamma\varepsilon^{1/8}$.

It was observed by Milman and Pajor [85] that all centroid bodies are zonoids;
that is, they can be obtained as limits, in Hausdorff distance, of a sequence
of convex bodies, called zonotopes, obtained as the Minkowski sums of finitely
many closed segments.

The concept of centroid body, using a different normalization, was extended to
$L_{p}$-Brunn-Minkowski theory by Lutwak and Zhang [74] in the following way.
For any real $p\geq 1$, let

$c_{d,p}=\frac{\kappa_{d+p}}{\kappa_{2}\kappa_{d}\kappa_{p-1}}.$

For any star body $K\subset\mathbb{R}^{d}$, let $\Gamma_{p}^{*}K$ be the
$o$-symmetric convex body defining the norm

$||x||_{\Gamma_{p}^{*}K}=\left(\frac{1}{c_{n,p}\operatorname{vol}_{d}(K)}\int_{K}|\langle
x,y\rangle|^{p}\,dy\right)^{1/p}.$

This body is called the _polar $L_{p}$-centroid body_ of $K$, and it can be
extended to the case $p=\infty$ by taking the limit $p\to\infty$. It is worth
noting that if $K$ is an $o$-symmetric convex body and $p=\infty$, then
$\Gamma_{p}^{*}K$ coincides with the Euclidean polar body $K^{*}$ of $K$,
defined as $K^{*}=\\{y:\langle x,y\rangle\leq 1\hbox{ for every }x\in K\\}$.
It was proved in [74] that for any star body $K$, we have

(2.7)
$\operatorname{vol}_{d}(K)\operatorname{vol}_{d}(\Gamma_{p}^{*}K)\leq\kappa_{d}^{2},$

with equality if and only if $K$ is an $o$-symmetric ellipsoid. This
inequality, for the case $p=\infty$, coincides with the Blaschke-Santaló
inequality. The polar body of $\Gamma_{p}^{*}K$ is called the _$L_{p}$
-centroid body of $K$_; if $p=1$, this body coincides with a normalization of
the centroid body $CK$ of $K$. We note that the body $\Gamma_{2}K$ also has a
geometric meaning: this is the ellipsoid which has the same inertia about
every axis as $K$. This body is called the _ellipsoid of inertia_ or _Legendre
ellipsoid_ of $K$ [15]. Verifying a conjecture in [74], Lutwak, Yang and Zhang
[72] proved that

(2.8) $\operatorname{vol}_{d}(K)\leq\operatorname{vol}_{d}(\Gamma_{p}K),$

with equality if and only if $K$ is an $o$-symmetric ellipsoid. Note that
(2.8), combined with the Blaschke-Santaló inequality, yields (2.7), and that
(2.8) is a generalization of the Busemann-Petty centroid inequality in (2.6).
A different proof of the inequality in (2.8) was given in [15], and a version,
establishing a stronger inequality for nonsymmetric bodies, can be found in
the paper [56] of Haberl and Schuster. The estimate in [56] was generalized
for compact sets in [61]. A different stronger version of the $L_{p}$
Busemann-Petty centroid inequality can be found in [76], and a variant for the
first quermassintegrals of the bodies in [113].

Let $\phi:\mathbb{R}\to[0,\infty)$ be a convex function satisfying $\phi(0)=0$
such that $\phi$ is strictly decreasing on $(-\infty,0]$ or strictly
increasing on $[0,\infty)$. The _Orlitz centroid body_ $\Gamma_{\phi}K$ of a
star body $K$ is defined as the convex body with support function

(2.9)
$h_{\Gamma_{\phi}K}(x)=\inf\left\\{\lambda>0:\frac{1}{\operatorname{vol}_{d}(K)}\int_{K}\phi\left(\frac{\langle
x,y\rangle}{\lambda}\right)\leq 1\right\\}.$

We note that if $\phi(t)=|t|^{p}$ with $p\geq 1$, then
$\Gamma_{\phi}K=\Gamma_{p}K$. This concept was introduced by Lutwak, Yang and
Zhang in [73], who proved that for any function $\phi$ satisfying the above
conditions, the ratio
$\frac{\operatorname{vol}_{d}(\Gamma_{\phi}K)}{\operatorname{vol}_{d}(K)}$ is
minimal on the family of convex bodies $K$ with $o\in\operatorname{int}(K)$ if
and only if $K$ is an $o$-symmetric ellipsoid. This result was extended for
arbitrary star bodies by Zhu [119], and the equality case, under a weaker
condition on $\phi$, was studied in [114].

It is an interesting question to find a converse inequality to (2.6). This
question was proposed by Bisztriczky and Böröczky Jr. [11] in 2001. They
proved that in the family of $o$-symmetric plane convex bodies (resp. plane
convex bodies containing $o$), the quantity
$\frac{\operatorname{area}(CK)}{\operatorname{area}(K)}$ is maximal if $K$ is
a parallelogram (resp. a triangle with $o$ as a vertex). Campi and Gronchi
[16] extended these results to $L_{p}$-centroid bodies in the plane, and Chen,
Zhou and Yang [19] to Orlitz centroid bodies in the plane.

The definition of Orlitz centroid bodies in (2.9) can be extended to that of
Orlitz-Lorenz centroid bodies by adding a weight function to the integral in
(2.9). The Busemann-Petty centroid inequality in this setting was investigated
in [91, 18, 42, 117, 116, 41]. Variants of centroid bodies, and the Busemann-
Petty centroid inequality, in non-Euclidean spaces (including complex,
spherical and hyperbolic spaces) can be found in [55, 68, 64, 8, 75].

##  3\. Static equilibrium points

Consider a convex body $K$ in $\mathbb{R}^{3}$. Clearly, if $K$ is balanced on
a horizontal plane $H$, then it is in static equilibrium if and only if the
orthogonal projection of its center of mass onto $H$ belongs to $K$. This
leads to the following definition (see e.g. [67]).

######  Definition 3.1.

Let $K$ be a convex body in $\mathbb{R}^{d}$ and $c\in\mathbb{R}^{d}$. A point
$q\in\operatorname{bd}(K)$ is called an _equilibrium point_ of $K$ _with
respect to $c$_ if the hyperplane passing through $q$ and perpendicular to
$[c,q]$ supports $K$.

We note that by convention, in Definition 3.1, every point
$c\in\operatorname{bd}(K)$ is an equilibrium point of $K$ with respect to $c$.
In the literature, $c$ is often assumed to be the centroid of $K$. In this
case we call $q$ simply an _equilibrium point_ of $K$. As we mentioned in the
beginning of Section 2, any reference point $c\in\operatorname{int}(K)$ is the
center of mass of $K$ equipped with some suitable, possibly inhomogeneous
density function. Thus, unless we state it otherwise, we assume from now on
that the reference point lies in the interior of the body.

For the case $c=o\in\operatorname{int}(K)$, a simple description of
equilibrium points can be given by means of the radial or the support function
of $K$. More specifically, if $\rho_{K}$ or $h_{K}$ is $C^{1}$-class, then $q$
is an equilibrium point of $K$ with respect to $o$ if and only if
$\frac{q}{|q|}$ is a critical point of $\rho_{K}$ or $h_{K}$, respectively.
Indeed, since the restriction of $\rho_{K}$ to a plane coincides with the
radial function of the intersection of $K$ with the plane, and a similar
observation holds for $h_{K}$ and the projection of $K$ onto the plane, it is
sufficient to show our remark for plane convex bodies. For this case a simple
geometric argument shows the statement (see also [31, 2]). We note that the
properties that $h_{K}$ or $\rho_{K}$ are $C^{1}$-class are not equivalent
[100], but if both are $C^{1}$-class, their critical points coincide.

A remarkable result of Zamfirescu [115] states that, in Baire category sense
with respect to the topology induced by Hausdorff distance, a typical convex
body has infinitely many equilibrium points with respect to a typical point.
On the other hand, Montejano [86] showed that the Euclidean balls centered at
$o$ are the only convex bodies whose every boundary point is an equilibrium
point with respect to $o$.

Next, we define non-degenerate equilibria.

######  Definition 3.2.

Let $K$ be a smooth convex body in $\mathbb{R}^{d}$ and $c\in\mathbb{R}^{d}$.
Assume that $K$ has finitely many equilibrium points with respect to $c$, and
its boundary is $C^{2}$-class in the neighborhood of each equilibrium point
$q\in\operatorname{bd}(K)$. Assume that the Hessian of the function
$\delta_{K}:\operatorname{bd}(K)\to\mathbb{R}$, $\delta_{K}(x)=|x-c|$ is not
zero at any equilibrium point $q$. Then we say that $K$ is _nondegenerate_
with respect to $c$. Furthermore, the _index_ of any equilibrium point is the
number of negative eigenvalues of the Hessian of $\delta_{K}$ at $q$.
Equilibrium points of index $0$ and $d-1$ are called _stable_ and _unstable_
points, respectively, and equilibrium points of all other indices are called
_saddle_ points.

We note that for the special case $c=o\in\operatorname{int}(K)$, there is an
equivalent formulation of the concepts in Definition 3.2 in terms of the
radial function $\rho_{K}$. Indeed, the conditions for the nondegeneracy of
$K$ are equivalent to the properties that $\rho_{K}$ is $C^{1}$-class, it has
finitely many critical points and $\rho_{K}$ is $C^{2}$-class in a
neighborhood of every one of them, and at every critical point the Hessian of
$\rho_{K}$ is not zero. Furthermore, the index of the equilibrium point
$q\in\operatorname{bd}(K)$ is equal to the number of the negative eigenvalues
of the Hessian of $\rho_{K}$ at the corresponding critical point
$\frac{q}{|q|}$.

Whereas there is no definition, in general, for a not necessarily smooth
convex body being nondegenerate, the condition in Definition 3.2 can be
replaced by a geometric one if $K$ is a polytope. We note that by definition,
every face (of any dimension) of a convex polytope contains at most one point
of equilibrium in its relative interior.

######  Definition 3.3.

Let $P$ be a convex polytope in $\mathbb{R}^{d}$, and let
$c\in\mathbb{R}^{d}$. For any equilibrium point $q$ of $P$ with respect to
$c$, let $H_{q}$ denote the supporting hyperlane of $P$ at $q$ perpendicular
to $[c,q]$. Set $F_{q}=P\cap H_{q}$, and let $f_{q}=\dim F_{q}$. If, for every
equilibrium point $q$, $q$ lies in the relative interior of $F_{q}$, we say
that $P$ is _nondegenerate_. In this case the _index_ of the equilibrium point
$q$ is the quantity $(d-1)-f_{q}$.

Consider a $C^{1}$-class function $f:\mathbb{S}^{d-1}\to\mathbb{R}$ with
finitely many critical points such that $f$ is $C^{2}$-class in a neighborhood
of every critical point, and its Hessian is nonzero at every critical point. A
standard convolution technique (see e.g. [48]) shows that there is a
$C^{\infty}$-class function $g:\mathbb{S}^{d-1}\to\mathbb{R}$ arbitrarily
close to $f$ such that for every $0\leq k\leq d-1$, the number of critical
points of $f$ with index $k$ coincide with the same quantity for $g$. The
function $g$ is a Morse function, and hence, applying this observation for the
radial function $\rho_{K}$ of a nondegenerate smooth convex body $K$ with
$c=o\in\operatorname{int}(K)$, and combining it with the Poincaré-Hopf formula
for the sphere $\mathbb{S}^{d-1}$, we have the following.

######  Theorem 3.4.

Let $K$ be a nondegenerate smooth convex body in $\mathbb{R}^{3}$ with respect
to $c\in\operatorname{int}(K)$, and let $S,H,U$ denote the numbers of stable,
saddle and unstable points of $K$ with respect to $c$. Then

(3.1) $S-H+U=2.$

Similarly, if $K$ is a nondegenerate plane convex body with respect to some
$c\in\operatorname{int}(K)$, with $S$ stable and $U$ unstable points, then
$S=U$.

Finally, let $P$ be a nondegenerate convex polytope in $\mathbb{R}^{d}$ with
respect to $c\in\operatorname{int}(P)$. Then, if $\varepsilon>0$ is
sufficiently small, the convex body $K=P+\varepsilon\mathbf{B}^{d}$ is a
nondegenerate convex body with respect to $c\in\operatorname{int}(K)$, and for
every $0\leq k\leq d-1$, the number of equilibrium points of $P$ of index $k$
coincide with the same quantity for $K$. Thus, Theorem 3.4 is satisfied also
for nondegenerate convex polyhedra and polygons. We also note that the notion
of radial function and Theorem 3.4 can be extended to the case
$c\notin\operatorname{int}(K)$ in a natural way [47].

###  3.1. Primary classes of convex bodies based on their equilibria

To distinguish convex bodies based on their static equilibrium properties it
is a straightforward way to use the numbers of their equilibrium points of
different indices. We observe that by Theorem 3.4, for any nondegenerate plane
convex body it is sufficient to know the number of either the stable or the
unstable points, and in $3$-dimensions two of the numbers of stable, saddle
and unstable points. Thus, we define the set $\\{S\\}$ as the family of all
nondegenerate convex bodies in $\mathbb{R}^{2}$ that have $S$ stable points,
and the set $\\{S,U\\}$ as the family of all nondegenerate convex bodies in
$\mathbb{R}^{3}$ that have $S$ stable and $U$ unstable points. For later use,
we decompose each class $\\{S\\}$ into two subclasses $\\{S\\}_{s}$,
$\\{S\\}_{p}$, consisting of the nondegenerate smooth plane convex bodies and
the convex polygons in this class, respectively, and we introduce the classes
$\\{S,U\\}_{s}$, $\\{S,U\\}_{p}$ in the same way (Figure 1). Elements of class
$\\{1,U\\}$ for some $U$ are called _monostable_ or _unistable_ , elements of
class $\\{S,1\\}$ are called _mono-unstable_ or _uni-unstable_ , and a body
that is both monostable and mono-unstable is called _mono-monostatic_.

htb

Figure 1. Equilibrium classes with some representative examples. Classes
corresponding to polyhedral pairs (see Section 3.6, below) are highlighted by
grey background. Unistable, uni-unstable, and mono-monostatic objects
correspond to the first row, the first column, and the top-left corner of the
table. The examples have been provided by Tímea Szabó and Gábor Domokos.

We call this classification the primary classification system, and a class
$\\{S\\}$ or $\\{S,U\\}$ the _primary class_ of its elements [111, 31]. It is
a natural question to ask what are the integers $S\geq 0$ (resp. the integers
$S,U\geq 0$) such that the class $\\{S\\}$ (resp. the class $\\{S,U\\}$) is
not empty. Our first step in answering this problem is the observation that if
$K$ is a nondegenerate convex body, then every boundary point of $K$ closest
to $c(K)$ is a stable, and every boundary point of $K$ farthest from $c(K)$ is
an unstable point of $K$. This implies that in every nonempty primary class,
$S,U$ are positive. The first nontrivial result in this direction is due to
Domokos, Papadopulos and Ruina [33], who proved that the class $\\{1\\}$ is
empty, or in other words, every nondegenerate plane convex body has at least
two stable and two unstable points. On the other hand, it is easy to see that
the class $\\{S\\}$ is not empty for any $S\geq 2$. Indeed, the class
$\\{2\\}$ contains all ellipses different from a circle, and for any $S\geq
3$, the class $\\{S\\}$ contains, e.g., the regular $S$-gons. This completes
the planar case. As a final remark, we mention that the result that $\\{1\\}$
is empty is equivalent to the famous $4$-vertex theorem in differential
geometry [111].

The characterization of nonempty primary classes $\\{S,U\\}$ was carried out
by Várkonyi and Domokos [111]. In particular, they presented a construction of
a convex body with only one stable and one unstable point. For preciseness, we
mention that the body constructed in [111] was not $C^{2}$-class at its two
equilibrium points, and thus, it is not nondegenerate as defined in Definition
3.2; nevertheless the radial function of the body has a global minimum and
maximum at its equilibrium points. We also add that, as it was remarked in
[31], a standard smoothing technique yields a $C^{\infty}$-class,
nondegenerate approximation of this body, with the same numbers of stable and
unstable points. For an explicit construction of nondegenerate convex bodies
in class $\\{1,1\\}$, with $C^{2}$-class boundaries, the reader is also
referred to [32]. A representative example of class $\\{1,1\\}$ constructed by
patching together elementary surfaces became known as ‘Gömböc’ (Figure 2). The
existence of a convex polyhedron in class $\\{1,1\\}$ was shown in [67].
Another remarkable simple shape in class $\\{1,1\\}$ given by an explicit
formula was recently published by Sloan [103], and possible symmetries of
mono-monostatic bodies have been determined in [32].

Once we have verified that the class $\\{1,1\\}$ is not empty, to show that no
class $\\{S,U\\}$, with $S,U\geq 1$ is empty it is sufficient to find a way to
increase the number of equilibrium points of a body. This was also carried out
in [111] by Domokos and Várkonyi, who showed that, in a neighborhood of a
fixed stable point of a convex body $K$, it is possible to create a stable
point by truncating $K$ by a suitable plane, or an unstable point by
truncating $K$ by a suitable cone. These steps are called ‘Columbus steps’ in
memory of the famous discoverer Christopher Columbus, who balanced an egg on
its top by creating a stable point near it. Since the class $\\{1,1\\}$ is not
empty, subsequent applications of these steps show that no class $\\{S,U\\}$
with $S,U\geq 1$ is empty. We note that a more rigorous investigation of
Columbus steps, involving also a construction of a smooth approximation of the
resulting body, can be found in [31]. The fact that all these classes contain
also nondegenerate convex polyhedra follows from [67].

Figure 2. Left: illustration of the ömböc: a piecewise smooth body in class
$\\{1,1\\}$. The boundary is coloured according to level curves of the radial
function, which has two critical points at the top and at the bottom of the
surface. Right: the monostatic polyhedron of Guy [21], see Section 3.6, below.
The illustration has been created with the aid of a 3D model available at
_https://www.thingiverse.com/thing:90866_.

###  3.2. Secondary an d tertiary classes

In this subsection, we investigate nondegenerate smooth convex bodies in
$\mathbb{R}^{3}$. Consider the radial function
$\rho_{K}:\mathbb{S}^{2}\to\mathbb{R}$ of such a body. By the smoothing
technique described in the beginning of Section 3, we may assume that
$\rho_{K}$ is a $C^{\infty}$-class function. Then $\rho_{K}$ is a Morse
function on $\mathbb{S}^{2}$, and the gradient of $\rho_{K}$ is nonzero at $q$
if and only if $q$ is not critical, i.e. it is regular. By the Picard-Lindelöf
theorem, for any regular point $q$ of $\mathbb{S}^{2}$ there is a unique curve
$\gamma:\mathbb{R}\to\mathbb{S}^{2}$ with the property that $\gamma(0)=q$, and
$\gamma^{\prime}(t)$ is equal to the gradient of $\rho_{K}$ at $\gamma(t)$ for
all $t\in\mathbb{R}$. Such a curve $\gamma$, called an _integral curve_ of the
gradient flow induced by $\rho_{K}$ starts and ends at two different critical
points of $\rho_{K}$, called the _origin_ and the _destination_ of $\gamma$.
For any critical point $x$, the _descending (resp. ascending) manifold_ of
$x$, denoted by $D(x)$ (resp. $A(x)$) is the union of $\\{x\\}$ and all
integral curves with $x$ as their destination (resp. origin). The connected
components of the nonempty sets, obtained as the intersection of an ascending
and a descending manifold, are called _cells_ of the _Morse-Smale complex
generated by $\rho_{K}$_. The $2$-, $1$-, and $0$-dimensional cells of this
complex are called _faces, edges_ , and _vertices_ , respectively.

If $x\neq y$ are critical points of $\rho_{K}$, then $A(x)\cap D(x)=\\{x\\}$,
and $A(x)\cap D(y)$ is the union of all integral curves with origin $x$ and
destination $y$. Thus, the vertices of the Morse-Smale complex of $\rho_{K}$
are the critical points of $\rho_{K}$, and if $x$ is a stable and $y$ is an
unstable point, then $A(x)\cap D(y)$ is an open set in $\mathbb{S}^{2}$,
implying that its connected components are faces of the complex (provided
${A(x)\cap D(y)\neq\emptyset}$). On the other hand, the same does not hold if
$x$ or $y$ is a saddle point since every saddle point is the origin, and also
the destination, of exactly two, _isolated_ integral curves, corresponding to
edges of the complex [39, 70].

The function $\rho_{K}$ is called _Morse-Smale_ if all ascending and
descending manifolds of $\rho_{K}$ intersect only transversally; or
equivalently, if any pair of intersecting ascending and descending
$1$-dimensional manifolds cross. In this case the crossing point of these
$1$-manifolds is necessarily a saddle, since crossing at a regular point would
contradict the property that any regular point belongs to exactly one integral
curve. In the following we only deal convex bodies whose radial function
$\rho_{K}$ is Morse-Smale. In this case every face of the Morse-Smale complex
is bounded by a union of lower dimensional cells, either two disjoint vertices
or a cycle of vertices and edges.

The existence of a face whose boundary is a pair of vertices implies that the
body $K$ has no saddle point, that is, its primary class is $\\{1,1\\}$. In
this case the Morse-Smale complex is made up of a unique face corresponding to
the single connected component of $A(x)\cap D(y)$, where $x$ is the stable and
$y$ is the unstable point of $K$, and its boundary consisting of the two
critical points $x$ and $y$. In addition, the function $\rho_{K}$ has no
saddle points, no isolated integral curves, therefore its Morse-Smale complex
has no edges. In the opposite case the boundary of every face consists of a
cycle of connecting edges and vertices, and it is known (see e.g. [39]) that
it consists of four edges, and the four critical points in the boundary are a
stable, a saddle, an unstable and a saddle point, in this cyclic order. The
boundary is possibly glued to itself along vertices and edges.

The topological graph $G$ on $\mathbb{S}^{2}$, whose vertices are the critical
points of $\rho_{K}$, and whose edges are the edges of the Morse-Smale
complex, is called the _Morse-Smale graph_ generated by $\rho_{K}$ [31]. This
graph is usually regarded as a _$3$ -colored_ quadrangulation of
$\mathbb{S}^{2}$, where the ‘colors’ of the vertices are the three types of a
critical point. We regard the topological graph $G$ defined by $\rho_{K}$ as
the drawing of a $3$-colored abstract graph $\bar{G}$ on $\mathbb{S}^{2}$. The
graphs $\bar{G}$ and $G$ are called the _secondary_ and the _tertiary class_
of $K$, respectively. The secondary classification of convex bodies is a
refinement of the primary one, as convex bodies whose Morse-Smale complexes
induce the same abstract graph have the same numbers of stable, saddle and
unstable points, whereas the statement is not necessarily true in the opposite
direction. Similarly, tertiary classification is a refinement of the secondary
one.

Let us denote by $\mathcal{Q}_{3}^{*}$ the family of finite $3$-colored
quadrangulations of $\mathbb{S}^{2}$, up to homeomorphism, where the color
classes are denoted by $\mathcal{S},\mathcal{H},\mathcal{U}$, each element of
$\mathcal{H}$ has degree $4$, and the cardinalities of the classes satisfy the
equation
$\operatorname{card}\mathcal{S}-\operatorname{card}\mathcal{H}+\operatorname{card}\mathcal{U}=2$.
We remark that the Morse-Smale complex of every Morse-Smale function on
$\mathbb{S}^{2}$ belongs to $\mathcal{Q}_{3}^{*}$. Furthermore, let us denote
the Morse-Smale graph of a nondegenerate convex body $K$, if it exists, by
$Q_{3}^{*}(K)$. We note that by convention, the Morse-Smale graphs of the
elements of the class $\\{1,1\\}$, consisting of two vertices with different
colors and no edges, are regarded as elements of $\mathcal{Q}_{3}^{*}$. These
graphs are called _path graphs_ on two vertices, and are denoted by $P_{2}$
(see Figure 3).

Figure 3. The topological graphs of convex bodies with at most two stable and
two unstable points.

As in the case of primary classes, it is a natural question to ask which
elements of$\mathcal{Q}_{3}^{*}$ appear as Morse-Smale graphs of nondegenerate
convex bodies. The answer, namely that _every_ such quadrangulation is
realizable by a convex body, was given in [31] by Domokos, Lángi and Szabó.
The idea of the proof is a refinement of Columbus steps, described in the
previous subsection. More specifically, a generalization of a method of
Batagelj [4], and also Negami and Nakamoto [90] shows that every $3$-colored
quadrangulation in $\mathcal{Q}_{3}^{*}$ can be reduced to $P_{2}$ by finitely
many applications of a graph operation, called _face contraction_ , or
equivalently, every such quadrangulation can be obtained from $P_{2}$ by
finitely many subsequent _vertex splittings_ , the dual operations of face
contractions. Based on this result, the authors of [31] show that for any
convex body $K$ and any $3$-colored quadrangulation $Q$ in
$\mathcal{Q}_{3}^{*}$, obtained from $Q_{3}^{*}(K)$ by an arbitrary vertex
splitting, there is a convex body $K^{\prime}$, obtained as a smoothened
truncation of $K$ by a plane or a cone, satisfying that
$Q_{3}^{*}(K^{\prime})=Q$.

We note that the above representation of the Morse-Smale complex of $K$ is
also called the _primal_ Morse-Smale graph of $K$. Saddle points can be
removed from the primal Morse-Smale graph without losing information: first we
connect maxima and minima in the quadrangles, then cancel saddle points and
edges incident to them (see Figure 4). This representation is called the
_quasi-dual_ Morse-Smale graph (see. [36]). Every face of a quasi-dual
representation is a quadrangle, and the vertices on its boundary form an
alternating sequence of two stable and two unstable points. Connecting the two
stable points in each face, and removing all unstable points and their edges
we obtain an even more compact representation of the Morse-Smale graph. This
representation is a planar graph that might contain loops and multiple edges.
The vertices of this graph are the stable points of the Morse-Smale complex,
and its edges correspond to the saddle points of the complex (See Figure 4).

It is a natural question to ask the numbers of secondary and tertiary classes
in each primary class. However, the answer to this question is known only for
some small values of $S$ and $U$.

Figure 4. Different representations of a Morse-Smale complex. a) Primal Morse-
Smale graph. b) Primal Morse-Smale graph with the stable and unstable points
connected. c) Quasi-dual Morse-Smale graph. d) Quasi-dual graph with the
stable points connected. e) The induced uncolored planar graph; vertices,
edges and faces correspond to the stable, saddle and unstable points of the
primal representation.

The definitions of the secondary and tertiary classes of a smooth
nondegenerate convex body $K$ rely on the topological properties of the Morse-
Smale complex of the gradient flow of the radial function $\rho_{K}$ of the
body. This task seems much more complicated if $K$ is a nondegenerate convex
polyhedron. Partial results in this direction can be found in the recent paper
[70] of Ludmány, Lángi and Domokos.

###  3.3. Transition between different classes

The next topic in our survey is to examine how difficult it is to move a
convex body from its class to another one. We start the investigation with
primary classes. Following the paper [26] of Domokos and Lángi, the question
we ask is this: For a given nondegenerate convex body $K$, what is the minimum
fraction of the volume of $K$ that is necessary to remove from $K$ to obtain a
nondegenerate convex body with a different number of equilibrium points? As we
have already seen in Subsection 3.1, this number is equal to zero for any
smooth convex body, because it is possible to use an arbitrarily small
truncation of $K$ to _increase_ the number of equilibrium points of the body.
This shows that we should aim at _decreasing_ the number of equilibrium points
of $K$. Thus, we define the _(downward) robustness_ of the convex body $K$ in
$\mathbb{R}^{d}$ as the quantity

$\rho(K)=\frac{1}{\operatorname{vol}_{d}(K)}\inf\left\\{\operatorname{vol}_{d}(K\setminus
K^{\prime}):K^{\prime}\in\mathcal{F}_{<}(K)\right\\},$

where the set $\mathcal{F}_{<}(K)$ denotes the family of convex bodies in
$K^{\prime}\subset K$ with fewer equilibrium points than $K$. Furthermore, we
define the robustness of an equilibrium class as the supremum of the
robustnesses of the elements of the class. To find the robustness of a convex
body, or an equilibrium class, is challenging partly because truncating a body
changes also the position of its centroid. This is the motivation behind
defining the _internal_ and _external robustness_ of a convex body. As in
[26], we do it for plane convex bodies with piecewise smooth boundaries, and
remark that the nondegeneracy of such a body, and its stable and unstable
points, can be defined for them by combining the conditions in Definitions 3.2
and 3.3. We denote the family containing such plane convex bodies by
$\mathcal{K}_{2}$.

######  Definition 3.5.

Let $K\in\mathcal{K}_{2}$ and $c\in\operatorname{int}(K)$. Assume that $K$ has
$S$ stable points with respect to $c$. Let
$R(K,c)\subseteq\operatorname{int}(K)$ denote the set of the points such that
$K$ has $S$ stable points with respect to any point of $R(K,c)$. The _internal
robustness_ of $K$ with respect to $c$ is

$\rho^{in}(K,c)=\frac{\inf\left\\{|q-c|:q\notin
R(K,c)\right\\}}{\operatorname{perim}(K)},$

where $\operatorname{perim}(K)$ is the perimeter of $K$.

The existence of the quantity $\rho^{in}(K,c)$ follows from the observations
that, first, for any $S>0$, the set with respect to which $K$ has $S$ stable
points is open, and second, Blaschke’s rolling ball theorem implies that any
$K\in\mathcal{K}_{2}$ contains two points with respect to which $K$ has
different numbers of stable points. The question of how the number of
equilibria depends on the location of the center of mass is discussed in
detail in Section 3.5, and the concept of internal robustness is illustrated
by Figure 5.

######  Definition 3.6.

Let $K\in\mathcal{K}_{2}$ and $c\in\operatorname{int}(K)$. Assume that $K$ has
$S$ stable points with respect to $c$. We define the _(downward) external
robustness of $K$_ with respect to $c$ as the quantity

$\rho^{ex}(K,c)=\frac{\inf\\{\operatorname{area}(K\setminus
K^{\prime}):K^{\prime}\in\mathcal{F}_{<}(K,c)\\}}{\operatorname{area}(K)},$

where $\mathcal{F}_{<}(K,c)$ denotes the family of the nondegenerate plane
convex bodies $K^{\prime}\in\mathcal{K}_{2}$, satisfying
$c\in\operatorname{int}(K^{\prime})\subset K$ , which has strictly less than
$S$ stable points with respect to $c$. If the set in the numerator is empty,
we let $\rho^{ex}(K,c)=1$.

The internal and the external robustness of the equilibrium class $\\{S\\}$ is
defined as the quantity
$\rho^{in}_{S}=\sup\\{\rho^{in}(K,c(K)):K\in\\{S\\}\\}$ and
$\rho^{ex}_{S}=\sup\\{\rho^{ex}(K,c(K)):K\in\\{S\\}\\}$, respectively.
Regarding these quantities, the authors of [26] prove that for any $S\geq 3$,
both $\rho^{in}(K,p)$ and $\rho^{ex}(K,p)$ are maximal over all
$K\in\mathcal{K}_{2}$ and $c\in\operatorname{int}(K)$ if $K$ is a regular
$S$-gon, and $c=c(K)$.

In [26], the authors extend the concepts of external and internal robustness
to nondegenerate smooth convex bodies and nondegenerate convex polyhedra in
$\mathbb{R}^{3}$ in the straightforward way, where, in case of internal
robustness, they replace the normalizing factor $\operatorname{perim}(K)$ by
$\sqrt{\operatorname{surf}(K)}$, where $\operatorname{surf}(\cdot)$ denotes
surface area. They show that any platonic solid in class $\\{S,U\\}$ has
maximal internal robustness among the elements of $(S,U)$ having the same
numbers of faces, edges and vertices, respectively. They also show that the
(full) robustness of the classes $\\{1,1\\}$, $\\{1,2\\}$ and $\\{2,1\\}$ is
equal to $1$.

Transitions between equilibrium classes can be described also from the point
of view of dynamical systems. The equivalence classes of nondegenerate, smooth
convex bodies are generic in the sense that a sufficiently small $C^{2}$-class
deformation of the body does not change any of its classes. In particular, if
$\\{K(t):t\in[0,1]\\}$ is a $C^{2}$-class differentiable $1$-parameter family
of convex bodies, where $K(0)$ and $K(1)$ belong to different equilibrium
classes, then there is some value $0<t_{0}<1$, $K(t_{0})$, where $K(t_{0})$ is
degenerate. If, in addition, for any $0\leq t<t_{0}$ (resp. $t_{0}<t\leq 1$)
$K(t)$ belongs to the same equilibrium class as $K(0)$ (resp. $K(1)$), then
there is a bifurcation of the corresponding gradient vector fields induced by
$\rho_{K(t)}$. In a ‘generic’ case, this bifurcation has codimension $1$ [54].
The codimension $1$ bifurcations of gradient vector fields on $\mathbb{S}^{2}$
have two types, called _saddle-node_ bifurcations and _saddle-saddle
connections_.

In terms of topological graphs, saddle-node bifurcations correspond to vertex
splittings/face contractions, while saddle-saddle connections correspond to
another type graph transformations on $3$-colored quadrangulations on
$\mathbb{S}^{2}$, called _diagonal slides_. These graph transformations
establish a combinatorial connection between primary/secondary or tertiary
equilibrium classes, namely we can say that the transition between two classes
$\mathcal{C}_{0}$ and $\mathcal{C}_{1}$ induced by such a graph transformation
$T$ is _combinatorially realizable_ if there are $3$-colored quadrangulations
$Q_{0}$ and $Q_{1}$, belonging to the classes $\mathcal{C}_{0}$ and
$\mathcal{C}_{1}$, respectively, such that $Q_{1}$ is obtained from $Q_{0}$ by
$Q_{1}$. In other words, we regard the $3$-colored quadrangulations of
$\mathbb{S}^{2}$, up to homeomorphism, as vertices of a ‘metagraph’
$\mathcal{G}$, and connect two vertices of $\mathcal{G}$ by an edge if one of
them is induced from the other one by a vertex splitting or a diagonal slide.
We call an edge _primary, secondary_ or _tertiary_ if the two quadrangulations
are induced by convex bodies in different primary, secondary or tertiary
classes, respectively.

It is a meaningful question to ask which edges of $\mathcal{G}$ are realizable
by a generic $1$-parameter family of convex bodies. This problem was partially
solved in [23], who proved that every primary and secondary edge of
$\mathcal{G}$ is realizable. The case of tertiary edges is still open.

###  3.4. The location and clustering of equilibria

Flattish objects like a coin tend to have two stable equilibrium points, and
elongated ones like a needle tend to have two unstable ones. This simple
observation can be turned into a more precise statement in various ways. For
example, it was shown in [28] that for any convex body $K$ in
$\mathbb{R}^{d}$, with $C^{\infty}$-class boundary and strictly positive
Gaussian curvature and for any orthogonal affinity
$f_{\lambda}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with affinity ratio $\lambda>0$,
if $\lambda$ is sufficiently large, then $f_{\lambda}(K)$ is exactly two
unstable points, and if $\lambda$ is sufficiently small, then $f_{\lambda}(K)$
has exactly two stable points.

Similarly, a convex body $K$ with a significantly flattened or elongated
_bounding box_ may not have equilibrium points but in some small regions of
$\operatorname{bd}K$ including two small patches and a narrow ring as
demonstrated by [34]. However, the number of equilibria within these areas can
be arbitrary. Indeed, many objects have localized clusters of equilibria. For
example, randomly generated fine polygonal/polyhedral discretizations of
smooth surfaces often possess small groups of nearby equilibria, in place of
isolated equilibria of the original surface. Based on this observation, global
equilibria consisting of clusters of local equilibria have been defined in
[34]. This concept was made precise in [30], where the authors gave simple
formulas for the numbers of stable, saddle and unstable points on a fine
polyhedral approximation of a $C^{2}$-class convex surface $S$ in
$\mathbb{R}^{3}$ induced by an equidistant partition of the parameter range,
based on the principal curvatures of the equilibrium points and their
distances from the reference points. They called these quantities the
_imaginary equilibrium indices_ of the equilibrium points of $S$, and carried
out a similar investigation of $C^{2}$-class curves in $\mathbb{R}^{2}$. In
[29], the authors removed some technical assumptions from the results in [30],
and proved a similar result for the number of equilibrium points of a
polygonal curve generated by randomly chosen points on the curve.

###  3.5. The number of equilibria with respect to a general reference point

Let $K$ be a convex body in $\mathbb{R}^{d}$. Let the function
$n_{K}(c):\mathbb{R}^{d}\to\mathbb{N}$ be defined as the number of equilibrium
points with respect to the point $c$. We note that if $K$ has $C^{2}$-class
boundary and strictly positive Gaussian curvature, then this quantity is equal
to the number of points of $\operatorname{bd}(K)$ through which the line
normal to $\operatorname{bd}(K)$ contains $c$. Thus, in the literature,
questions related to this function are often referred to as _problems of
concurrent normals_. In this subsection we discuss results related to
$n_{K}(\cdot)$.

First, we consider plane convex bodies. The function $n_{K}(\cdot)$ was first
investigated by Apollonius of Perga [92] in the special case of an ellipse,
and found that its value in a generic point is either 2 or 4. It is known that
for any $K$ in $\mathbb{R}^{2}$ with $C^{2}$-class boundary and strictly
positive curvature, $n_{K}(\cdot)$ is a piecewise constant function. Even
more, it was proved by Allemann, Hungerbühler and Wasem [2] that
$n_{K}(c)=2+2|m|$ where $m$ is the winding number of the evolute of
$\operatorname{bd}(K)$ with respect to $c$. This result makes it possible to
have a nice illustration of the function $n_{K}(\cdot)$: we decompose the
plane into connected regions by the evolute of $\operatorname{bd}(K)$ (which
in the following we call the evolute of $K$) and observe that within each
region $n_{K}(\cdot)$ is a constant (Figure 5). As a consequence, $n_{K}(c)$
is discontinuous at the points of the evolute, and $K$ is degenerate only with
respect to the points of its evolute.

It is trivial that the number of equilibrium points of any convex body $K$
with respect to any point $c$ is at least two; this follows from the
observation that the points of $\operatorname{bd}(K)$ closest to and farthest
from $c$ are equilibrium points. On the other hand, for any convex body with
$C^{2}$-class boundary and strictly positive Gaussian curvature, $n_{K}(c)=2$
for all points $c$ sufficiently far from $K$. However, due to motivation by
mechanics, it is often assumed that the reference point $c$ lies in the
interior of the convex body $K$. Thus, we define the quantity

$n_{min,K}=\min\\{n_{K}(c):c\in\operatorname{int}(K)\\}.$

We observe that there are plane convex bodies $K$ satisfying $n_{min,K}\geq
4$. A simple example for this is a smoothened regular triangle (Figure 5).
Even higher values of $n_{min,K}$ are possible if we drop the
differentiability properties for $\operatorname{bd}(K)$: for example,
rectangles have eight equilibrium points with respect to all interior points
or more generally, bricks in $\mathbb{R}^{d}$ have $3^{d}-1$ equilibrium
points with respect to all their interior points. The highest possible value
of $n_{min,K}$ among polytopes and among smooth bodies appear to be unsolved
problems. For example, it is an intriguing question if there exist a plane
convex body $K$ with $C^{2}$-class boundary and strictly positive curvature
that satisfies $n_{min,K}>4$. A related result by Grebennikov [51] is that
almost every normal through a boundary point to such a convex body in any
dimensions contains a point $c$ with $n_{K}(c)\geq 6$.

Figure 5. Two examples of smooth, planar curves with their evolutes, and the
corresponding number of equilibria. In the left panel, the dashed circle
illustrates the internal robustness of a point P. The right one is an example
with $n_{min,K}=4$.

In contrast, the quantity

$n_{max,K}=\max\\{n_{K}(c):c\in\operatorname{int}(K)\\}$

has been studied by several authors. More specifically, it has been proven
under mild smoothness assumptions that $n_{max,K}\geq 2d$ for any convex body
with $C^{1}$-class boundary in $\mathbb{R}^{d}$ with $2\leq d\leq 4$ [59, 94],
and the same statement in arbitrary dimensions has been known for long as
_concurrent normal conjecture_. In addition, the inequality $n_{max,K}\geq 6$
has been proven for any convex body $K$ in $\mathbb{R}^{d}$ with an arbitrary
value of $d$ [59]. In a recent paper, another lower bound for $n_{max,K}$,
over the family of $d$-dimensional convex polytopes, was given by Nasonov,
Panina and Siersma[89], namely they proved the inequality $n_{max,K}\geq
2d+2$, and conjectured that the inequality $n_{max,K}\geq 10$ holds for any
$3$-dimensional convex polytope.

Several works have addressed the average number of equilibria

(3.2)
$\displaystyle\overline{n}_{K}=\frac{1}{\operatorname{vol}_{d}(K)}\int_{\operatorname{int}(K)}n_{K}(x)dx$

in a convex body $K$, for certain families of convex bodies. A list of bounds,
established in the papers [27, 57, 62, 99, 58, 37], have been collected in
Table 1.

Dimension | 2 | 3 | $d$
---|---|---|---
$\mathcal{P}$ |  $\stackrel{{\scriptstyle?}}{{>}}4$ [37], $\leq 12$ [57] |  $\stackrel{{\scriptstyle*}}{{>}}2$ [27] |  $\stackrel{{\scriptstyle*}}{{>}}2$[27]
$\mathcal{F}$ |  $\stackrel{{\scriptstyle*}}{{\geq}}2$, | $\stackrel{{\scriptstyle*}}{{\geq}}2$ | $\stackrel{{\scriptstyle*}}{{\geq}}2$
|  $\leq 12$ [57, 62]; $\stackrel{{\scriptstyle?}}{{\leq}}8$ [58] |  $\leq 62$; $\stackrel{{\scriptstyle?}}{{\leq}}26$ [58] | $\leq\frac{3}{2}^{d}\binom{2d}{d}-1$
$\mathcal{F}_{cs}$ |  $\stackrel{{\scriptstyle*}}{{\leq}}8$ [57, 62] |  $\stackrel{{\scriptstyle*}}{{\leq}}26$ [57, 62] |  $\stackrel{{\scriptstyle*}}{{\leq}}3^{d}-1$ [57, 62]
$\mathcal{F}_{cc}$ |  $\leq 6$ [57, 62] |  $\leq 20$ [27] |  $\leq\binom{2d}{d}$ [27]
$\mathcal{F}_{cc}$ |  $\stackrel{{\scriptstyle*}}{{\leq}}\frac{2\pi}{\pi-\sqrt{3}}$ [99, 17] |  $\leq 20$ [27] |  $\leq\binom{2d}{d}$ [27]
Table 1. A summary of known bounds of the average number of equilibria. Star
means bounds verified to be sharp, and question marks denote published
conjectures. The notation in the first row is as follows: $\mathcal{P}$ and
$\mathcal{F}$ are the families of convex polytopes, convex bodies with
$C^{2}$-class boundary and strictly positive curvature, respectively, in the
appropriate dimension, and for $x\in\\{cs,cc,cw\\}$, $\mathcal{F}_{x}$ denotes
the subfamily of $\mathcal{F}$ consisting of centrally symmetric convex
bodies, of the convex bodies containing all their centers of curvature, and of
constant width bodies. The bounds established by [62] do not require
$C^{2}$-class boundary.

###  3.6. Equilibrium points of convex polyhedra

In modern times, the study of equilibrium points of convex polyhedra was
probably started with a problem of Conway and Guy [21] in 1966 who conjectured
that there is no monostable tetrahedron but there is a monostable convex
polyhedron in $\mathbb{R}^{3}$. These two questions were answered by Goldberg
and Guy in [50] in 1969 (Figure 2). For a more detailed proof of the first
problem, see also [77]. In addition, in [50] Guy presented some problems
regarding monostable polyhedra, stating that three of them are due to Conway
(for similar statements in the literature, see e.g. [22, 80, 38]). These three
questions appear also in the problem collection of Croft, Falconer and Guy
[22] as Problem B12. These problems were:

  1. (1)

Can a monostable polyhedron in the Euclidean $3$-space $\mathbb{R}^{3}$ have
an $n$-fold axis of symmetry for $n>2$?

  2. (2)

What is the smallest possible ratio of diameter to girth for a monostable
polyhedron?

  3. (3)

What is the set of convex bodies uniformly approximable by monostable
polyhedra, and does this contain the sphere?

Related to these problems we recall that the _girth_ of a convex body in
$\mathbb{R}^{3}$ is the minimum perimeter of an orthogonal projection of the
body onto a plane [38]. It is worth noting that, as stated in [50], Conway
showed that no body of revolution can be monostable, and also that the
polyhedron constructed in [50] has a $2$-fold rotational symmetry.

The first two questions have recently been answered by Lángi [67], but the
third one is still open. The main tool of the proofs of the result in [67] is
a theorem stating that a nondegenerate convex body, under some mild condition
on its curvatures, can be approximated arbitrarily well by a nondegenerate
convex polyhedron with the same numbers of equilibria and with the same
symmetry group. This result can be regarded as the counterpoint of the one in
[30] about approximations of a convex surface by equidistant partitions of its
parameter range, mentioned in Subsection 3.4.

A problem proposed in [50] asks about the minimum dimension, if it exists, in
which a $d$-simplex can be monostable. This problem has been investigated by
Dawson et al. [77, 79, 78, 80], who proved that there is no monostable
$d$-simplex if $d\leq 8$ and there is a monostable $11$-simplex. With regard
to Problem XVI in [101], asking about the minimum number of faces of a
monostable polyhedron in $\mathbb{R}^{3}$, the original construction of Guy
[50] with $19$ faces (attributed also to Conway) was modified by Bezdek to
obtain a monostable polyhedron with $18$ faces [9], while a computer-aided
search by Reshetov [97] yields a monostable polyhedron with $14$ faces. The
non-existence of mono-unstable homogeneous tetrahedra has been proved by [25]
where a construction of a mono-unstable polyhedron was also given.

We have seen that all equilibrium classes $\\{S,U\\}$ contain nondegenerate
convex polyhedra [67] and all classes $\\{S\\}$ contain convex polygons [67],
as one can construct a fine polyhedral appoximation of a smooth convex body
from the same class. On the other hand, these representatives have much more
faces, edges, and vertices than the number of equilibria. Recall that every
vertex, edge or face of a nondegenerate convex polyhedron contains at most one
equilibrium point in its relative interior. Thus, an interesting related
problem is to find the ‘simplest’ convex polyhedron in each class in terms of
the number of vertices, edges or faces. This leads to the following concept.

For an arbitrary nondegenerate, convex polyhedron $P$ the total number of
faces, edges, and vertices without equilibria is called the _complexity_ of
$P$, denoted by $C(P)$. Furthermore, for any $S,U\geq 1$, we denote the
smallest complexity among polyhedra in the class $\\{S,U\\}_{p}$ by $C(S,U)$,
and call it the _complexity of equilibrium class $\\{S,U\\}_{p}$ _. In
addition, recall a classical theorem of Steinitz [105, 104] stating that a
triple of integers $(v,e,f)$ are the numbers of the vertices, edges and faces
of a convex polyhedron, respectively, if, and only if $f\geq 4$,
$\frac{f}{2}+2\leq v\leq 2f-4$, and $e=v+f-2$. Based on this result, let us
call a pair of integers $(f,v)$ a _polyhedral pair_ if they satisfy these
inequalities, and let $\mathcal{PP}$ denote the set of all polyhedral pairs.

It has been shown in [25] that for all equilibrium classes with $S,U>1$,

$C(S,U)=2\min\\{f+v-S-U:(f,v)\in\mathcal{PP}\\}.$

In particular, this result implies that $C(S,U)=0$ if and only if $(S,U)$ is a
polyhedral pair, or in other words, any primary equilibrium class
$\\{S,U\\}_{p}$, where $(S,U)$ is a polyhedral pair, can be represented by a
_minimal polyhedron_ , that is, by a polyhedron that contains an equilibrium
point in the relative interior of every face, edge and vertex. An illustration
of polyhedral pairs, and some examples of minimal polyhedra are given in
Figure 1. There is no similar result for secondary and tertiary equilibrium
classes, though the problem seems to be related to the problem of centering
Koebe polyhedra (see [25], and for more information on centering Koebe
polyhedra, see [66]).

While the above result answers our question for all classes except for
monostable and mono-unstable ones, only estimates are known about the
complexities of the classes of monostatic polyhedra. In particular, the
monostable polyhedron $P_{CG}$ of Conway and Guy in [21] is in class
$\\{1,4\\}_{p}$ and has complexity $C(P_{CG})=96$. The ones $P_{B}$ and
$P_{R}$, constructed by Bezdek and Reshetov, are in classes $\\{1,3\\}_{p}$
and $\\{1,2\\}_{p}$, respectively, having complexity $C(P_{B})=64$ and
$C(P_{R})=70$. The authors of [25] extended these constructions for all
classes $\\{1,U\\}_{p}$ with $U\geq 2$, and constructed polyhedra in classes
$\\{S,1\\}_{p}$ for all $S\geq 2$, yielding the estimates $C(2,1)\leq 66$,
$C(3,1)\leq 64$, and for all $S,U\geq 4$, $C(S,1)\leq
61+2\left\lfloor\frac{S}{2}\right\rfloor$ and $C(1,U)\leq
92+2\left\lfloor\frac{U}{2}\right\rfloor$. A lower bound $v\geq 7$ for the
number $v$ of vertices of mono-unstable polyhedra was established by [14]
using a computational approach based on semi-definite programming. Presently
there is no explicit estimate for the complexity of the class $\\{1,1\\}_{p}$.
However it has been conjectured that $C(1,1)$ is at least approximately $1000$
[111].

The picture outlined above changes significantly, if the centroid of the
polyhedron is replaced by the center of mass of the vertices. In this case,
mono-unstable polyhedra with less than eight vertices do not exist [13], but a
mono-monostatic polyhedron with $21$ faces and $21$ vertices has been found
[24]. If the assumption of homogeneity is relaxed, and the reference point is
permitted to be an arbitrary interior point of the polyhedron, then even some
(non-regular) tetrahedra may be monostable as first pointed out by Conway (see
[80]) whereas other tetrahedra may be mono-unstable [3]. It is also shown in
[3] that all monostable and mono-unstable tetrahedra belong to the equilibrium
class $\\{1,2\\}_{p}$ and $\\{2,1\\}_{p}$, respectively. In higher dimensions,
even some regular polytopes may be monostable [80].

##  4\. Related problems and applications

The concepts of equilibrium and stability are closely related to the
equilibria of floating bodies, which is the main focus of a review paper of
Alfonseca et al. in preparation. The equilibrium positions of a convex body
$K$ floating in a liquid correspond to the equilibrium positions in the sense
of Definition 3.1 of a related body $U_{V}(K)$, called the _Ulam floating
body_ of $K$ [60]. The boundary of this body is also referred to as the
_surface of buoyancy_ (Alfonseca et al., in preparation). For any
$0<V<\operatorname{vol}_{d}(K)$, the surface of buoyancy contains the
centroids of the intersections of $K$ with all closed half spaces $H$ that
satisfy the condition $\operatorname{vol}_{d}(K\cap H)=V$. Thus, if $K$ is
symmetric and $V=\frac{1}{2}\operatorname{vol}_{d}(K)$ coincides with the
centroid body of $K$ (see Subsection 2.2). Moreover, in the limit cases $V\to
0$ and $V\to\operatorname{vol}_{d}(K)$, up to homothety, $U_{V}(K)$ tends to
the original body $K$.

A possible generalization of the notion of equilibrium point of a convex body
is the following [2]: given an angle $0<\alpha<\frac{\pi}{2}$, we say that a
point $q$ in the boundary of a convex body $K$ is an _equilibrium point of $K$
of parameter $\alpha$_, with respect to $c$, if $K$ has a supporting
hyperplane at $q$ with angle equal to $\frac{\pi}{2}-\alpha$ to $[c,q]$. This
definition is motivated by physical objects that can be balanced at a point on
planes with slope angle $-\frac{\pi}{2}<\alpha<\frac{\pi}{2}$ provided that
there is sufficient friction to prevent downhill slip. The extension of
stability is also straightforward in two dimensions in the case $c=o$ by
noting that such points correspond to critical points of the function
$u\mapsto\cos\alpha h_{K}(u)+\sin\alpha s(u)$, $u\in\mathbb{S}^{1}$, where
$s(u)$ is the (signed) arclength of the arc of $\operatorname{bd}(K)$ between
$\rho_{K}(u)u\in\operatorname{bd}(K)$ and an arbitrary fixed reference point
in $\operatorname{bd}(K)$. This function corresponds to the potential energy
function of a heavy plane body rolling on a slope without slip. Allemann,
Hungerbühler and Wasem [2] showed that the number of equilibria of parameter
$\alpha$ of a plane convex body $K$ is a piecewise constant function of
$\alpha$, which drops to zero outside a bounded interval (with 0 in its
interior) of the slope angle $\alpha$. In dimensions $d\geq 3$ there is no
straightforward way of defining stability, as strictly convex bodies can
always escape from an equilibrium by means of a spin motion.

The concepts outlined in this paper can also be extended to equilibria of
objects enclosed in a sphere of radius $\alpha$ [110]; in this context the
equilibrium points in Definition 3.1 correspond to the limit case
$\alpha\to\infty$. The equilibria of a convex body $K$ in a sphere of radius
$\alpha$ are in one-to-one correspondence with the equilibrium points of the
$\alpha$-hull of $K$, defined as the intersection of all balls of radius
$\alpha$ that contain $K$ (for the definition of $\alpha$-convex sets and a
systematic investigation of these bodies, the reader is referred to [83] and
[10], respectively). The number of equilibrium points of a plane convex body
$K$ with respect to an interior point $c$, defined in this way, is a piecewise
constant, strictly increasing function of $\alpha$, with minimum value of two
for every $c$ different from the center of the smallest ball containing $K$.
In case of $3$-dimensional convex bodies, this function is not necessarily
monotonic, but it also drops to two if $\alpha$ is sufficiently close to the
circumradius of the body. In other words, almost all objects become mono-
monostatic inside a sufficiently small sphere [110].

A further generalization considers equilibria of objects with multiple contact
points on an underlying surface with arbitrary geometry. A partial
characterization of the positions of the centers of mass corresponding to
equilibria has been given by [93, 98], and rich geometric structures have been
uncovered. The stability analysis of these systems relies heavily on the
theory of dynamical systems [96, 112] and is beyond the scope of this work.

We finish the paper by briefly mentioning a few applications of the above
results outside mathematics. The theorem of Winternitz (see Subsection 2.1)
has recently found an application to data depth in statistics [88].
Equilibrium classes (see Subsections 3.1, 3.2) are used as a shape
classification scheme in geology [65, 34, 69]. Flocks of equilibria (see
Subsection 3.4) help to understand fascinating geological formations called
_balancing rocks_ [5, 71].

Monostable objects play special roles in industrial object manipulation
processes called part feeding [6], and the statistical behaviour of randomly
dropped objects with multiple stable equilibria is also attracts much interest
[49, 109]. Monostable shapes also appear in nature [35, 20] as well as in
robotics [108, 118] , and they allow spontaneous self-righting if an object
topples. A further medical application of monostable objects concerns
autonomous drug delivery [1].

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