# Temporal higher-order interactions facilitate the evolution of cooperation

Xiaochen Wang,1,† Lei Zhou,2,† Zhenglong Tian,1 and Aming Li1,3,∗ 1. Center
for Systems and Control, College of Engineering, Peking University, Beijing
100871, China 2. School of Automation, Beijing Institute of Technology,
Beijing 100081, China 3. Center for Multi-Agent Research, Institute for
Artificial Intelligence, Peking University, Beijing 100871, China ${\dagger}$
These authors contributed equally to this work $*$ Corresponding author.
E-mail<EMAIL_ADDRESS>
###### Abstract

Motivated by the vital progress of modeling higher-order interactions by
hypernetworks, where a link connects more than two individuals, we study the
evolution of cooperation on temporal hypernetworks. We find that temporal
hypernetworks may promote cooperation compared with their static counterparts.
Our results offer new insights into the impact of network temporality in
higher-order interactions on understanding the evolution of cooperation,
suggesting traditional networks based on pairwise or static interactions may
underestimate the potential of local interactions to foster cooperation.

††preprint: APS/123-QED

##  I Introduction

Higher-order interactions, which occur among multiple individuals, are
prevalent in complex systems, such as co-authorship in academia [1],
mesoscopic interactions among neurons in the nervous system [2], and
competitions among species in ecosystems [3]. When these interactions give
rise to social dilemmas, cooperation, which entails a personal cost and
benefits others, is supposed to be outperformed by defection which free-rides
in terms of both individual rationality and Darwin’s notion of ‘survival of
the fittest’ [4]. However, cooperation evolves and persists in nature [5, 6,
7, 8, 9, 10]. Network reciprocity is one of the fundamental explanations for
the evolution of cooperation [11], where networks impose local interactions
and enable cooperators to form clusters to avoid exploitations by defectors
[10, 12, 13, 14, 8, 9, 11, 15, 16, 17].

Most previous studies about higher-order interactions on networks focus on
traditional pairwise network models, where nodes represent individuals and
links interactions [18]. Following this framework, a widely adopted setting
for capturing higher-order interactions is that each individual and all its
neighbors form a group and interact together [12, 19, 18]. This means
individuals interact with not only their neighbors but also their neighbors’
neighbors. However, while changing behaviors, individuals can only imitate
their neighbors rather than all interaction partners [12, 20], suggesting an
incompatible definition of groups and the incapability of traditional networks
to accurately describe higher-order interactions [16, 21]. Recent works point
out that hypernetworks that use hyperlinks (i.e., sets of nodes, or
generalized versions of ‘links’ that can connect more than two individuals) to
describe groups are better and more natural network models to capture higher-
order interactions than traditional networks [22, 23]. And it is shown that
the presence of higher-order interactions, compared with pairwise interactions
on traditional networks, has a fundamental impact on the evolution of
cooperation [22].

Although hypernetworks provide a valid tool for studying the effects of
higher-order interactions on the evolution of cooperation, current studies
assume that these interactions exist permanently and never change. In reality,
interactions among individuals build and dissolve over time, indicating that
links only exist intermittently. For example, human interactions through
digital or face-to-face communications usually last finitely in short
durations [24] and ecological networks alter with the seasons [25, 26]. More
importantly, it has been shown that such network temporality can significantly
alter various dynamical processes on networks, such as the spreading of
information [27], the evolution of cooperation [28], and controllability of
systems [20].

Here, we systematically investigate the evolution of cooperation on temporal
hypernetworks, where hypernetwork structures change over time. By combining
numerical simulations and theoretical analysis on both synthetic and empirical
temporal networks, we find that temporal hypernetworks with higher-order
interactions and sparse components can strongly promote cooperation, and even
outperform their static counterparts. Our results thus shed new light on the
effect of temporal higher-order interactions on the evolution of cooperation.

Figure 1: The construction of temporal hypernetworks. a. Social interactions
between 4 individuals are indicated by solid circles with different colors.
Each individual is depicted by the line, over which the corresponding circles
will be connected at time $t$, provided two players interact with each other
during the time interval $(t-1,t]$. b. The snapshots are generated from all
interactions during the window $\Delta t$. Several individuals are seen in a
hyperlink if they interact with each other during the window $(t-\Delta t,t]$.
c. The snapshots within each time interval $\tau$ are then aggregated into
subhypernetworks. For example, the green and yellow nodes interact on the
first and second snapshots, thus there is a hyperlink between them with weight
2 on the first subhypernetwork. d. All snapshots are aggregated into the
static hypergraph.

##  II Model

Here, we model a temporal hypernetwork as a series of subhypernetworks. On
each subhypernetwork, a hyperlink connects multiple individuals. As
subhypernetworks change over time, hyperlinks can build or dissolve. We first
construct empirical temporal hypernetworks using empirical datasets. The
dataset can be modeled as a sequence of $\mathcal{G}$ snapshots, where each
contains all interactions during a non-overlapping time window $\Delta t$ on a
fixed population of $N$ individuals (Fig. 1a). In each snapshot, every clique
is represented by a hyperlink, in which individuals interact with each other
(Fig. 1b). The snapshots within each time interval $\tau$ are then
subsequently aggregated, generating a sequence of
$\lceil\mathcal{G}/(\tau/\Delta t)\rceil$ subhypernetworks that form a
temporal hypernetwork (Fig. 1c). When $\tau=\mathcal{G}\Delta t$, we obtain
the corresponding static hypernetwork (Fig. 1d).

Moreover, we also construct synthetic temporal hypernetworks by using the
Holme-Kim model [29], which generates a static network. To construct each
snapshot for the synthetic temporal hypernetwork, the links of the static
network are activated with a probability $p$, and the resulting cliques are
represented by hyperlinks. This process is repeated $\mathcal{G}$ times to
construct each snapshot. After that, a series of synthetic subhypernetwokrs
are constructed and form the synthetic temporal hypernetwork via the above
procedure introduced for the empirical temporal hypernetworks.

On temporal hypernetworks, individuals within the same hyperlink play the
public goods game, where each cooperator ($C$) pays a cost $c$, and each
defector ($D$) pays nothing. The total costs are then collected and multiplied
by a synergy factor $r$ to generate payoffs that are evenly distributed to all
individuals within the hyperlink. The payoff of each defector and cooperator
is

$\pi_{D}(n_{C})=rcn_{C},$

and

$\pi_{C}(n_{C})=rcn_{C}-c,$

where $n_{C}$ is the number of cooperators within the hyperlink. In each
round, each individual $i$ accumulates the payoff $P_{i}$ from all games she
joins , namely, all hyperlinks she belongs to on the subhypernetwork.

When it turns to strategy updating, all individuals decide whether to change
their strategies or not. Individual $i$ selects a random group and compares
her payoff with a random neighbor $j$ in the group. Here individual $i$
imitates $j$’s strategy with probability given by the Fermi function
$1/\\{1+\exp[-s({P}_{j}-{P}_{i})]\\}$, and $s$ is the intensity of selection.
Individuals play games with their neighbors $g$ rounds on every
subhypernetwork before it changes to the next one.

To tell whether the evolution of cooperation is promoted, we focus on the
critical value $r_{C}$, where the frequency of cooperators is starting to be
above zero. A smaller $r_{C}$ means that cooperation is easier to emerge and
the corresponding model settings (e.g., parameters, network structures) are
said to better promote the evolution of cooperation.

##  III Results

###  III.1 Higher-order interactions and sparse components promote cooperation

Figure 2a shows that when the time interval $\tau$ of the subhypernetworks is
constant, an increase in the time window $\Delta t$ significantly promotes the
frequency of the cooperators $f_{C}$ and decreases the critical values
$r_{C}$. For a fixed $\Delta t$, as shown in Fig. 2b, cooperation is
facilitated when a subhypernetwork contains interactions within a moderate
interval (i.e., moderate $\tau$). Intuitively, the window $\Delta t$ and
interval $\tau$ shape the temporal hypernetwork series. Enlarging the window
will allow more interactions to be encapsulated in the snapshots, making the
sizes of hyperlinks larger and the degree (i.e., the number of hyperlinks each
individual belongs to) higher. And large $\tau$ can increase the degree and
size of components on subhypernetworks, which will affect the aggregation of
cooperators.

Figure 2: Higher-order interactions and sparse components promote cooperation.
We plot the frequency of cooperators $f_{C}$ as a function of the synergy
factor $r$ on empirical (a, b) and synthetic (f, i) temporal hypernetworks.
For synthetic hypernetwork, we set the probability for links to be activated
$p$ to be $0.05$ (f) and $0.6$ (i). The cooperation emerges at a smaller
$r_{C}$ for a larger $\Delta t$ and intermediate $\tau$. c. We give an example
of the definition of connected density. For the two networks with 4 nodes and
2 links, the traditional density is $\rho^{\prime}=1/3$. However, connected
density focuses on the components. In the top network, there are two
components with densities equal to 1. Then the connected density of the whole
network is 1. For the other network (bottom) with one component and one
isolated node, it has a connected density of 1/2. We present the average
connected density $\langle\rho\rangle$ as a function of time interval $\tau$
of empirical (d) and synthetic (g, j) hypernetworks. The unit of $\tau$ is
1min for panel d. For empirical and synthetic hypernetwork with $p=0.05$, the
connected density drops sharply and then increases slowly when $\tau$
increases. However, it increases monotonically on synthetic hypernetworks with
$p=0.6$. And we plot the average hyperlink size $\langle n\rangle$ as a
function of window $\Delta t$ of the empirical hypernetwork (e). The size
increases as $\Delta t$ goes up. We also display distributions of the
hyperlink size for synthetic hypernetwork (f, k). The synthetic hypernetworks
with $p=0.6$ have a higher size compared with the hypernetworks with $p=0.05$.
For the statics, the last subhypernetwork is ignored if $\tau\nmid G\Delta t$.
The simulations are performed $10^{3}$ times, and within each time, the
evolution is repeated $10^{6}$ rounds. Every independent simulation starts
with an equal number of cooperators and defectors. For all simulations, the
round of interaction $g=100$. The population size $N=327$ for empirical
hypernetworks and $N=100$ for synthetic hypernetworks. We set $\tau=5$h for
pannel a, and $\Delta t=0.1$h for pannel b.

For an intuitive explanation, we consider a degree-homogeneous uniform
temporal hypernetworks, in which each individual has the same degree and all
hyperlinks have the same size $n$ in every subhypernetwork. We theoretically
approximate the critical value $r_{C}$ for cooperators to emerge using the
replicator equation,

$\displaystyle\dot{x}_{D}=$
$\displaystyle(1-\delta)\sum_{i=0}^{n-1}\binom{n}{i}\left(1-x_{D}\right){}^{i}x_{D}^{n-i}\frac{i(n-i)}{n(n-1)}f(\pi_{D}-\pi_{C})$
$\displaystyle+\delta\sum_{i=0}^{n-1}\binom{n}{i}p^{i}(1-p)^{n-i}\frac{i(n-i)}{n(n-1)}f(\pi_{D}-\pi_{C}),$
(1)

where $x_{D}$ is the frequency of defectors, and
$f(\pi_{D}-\pi_{C})={1}/\\{1+\exp[-s(\pi_{D}-\pi_{C})]\\}-{1}/\\{1+\exp[-s(\pi_{C}-\pi_{D})]\\}.$
capturing the net growth rate of cooperators. Parameter $\delta$ is the
similarity between consecutive subhypernetworks. As subhypernetworks switch
every step, $1-\delta$ of the hyperlinks are regenerated. $p$ is the
probability that an individual is a defector on the unchanged hyperlinks (see
Supplemental Material for details). Solving $\dot{x}_{D}=0$, we have

$r_{C}=\frac{1}{1+(n-1)(1-2q)\delta},$ (2)

where $q<1/2$ is a parameter related to the probability that neighbors are
using different strategies during the evolution on the unchanged hyperlinks.
As shown in Eq. (2), we demonstrate that a larger group (hyperlink) size $n$
reduces the critical value $r_{C}$, and thus promotes cooperation.

Previous studies indicate that a sparser network leads to a more pronounced
aggregation phenomenon, representing a smaller $q$ [30, 31]. To better
represent the connectivity between nodes, we use the concept of network
density. First, we map the hypernetwork to an unweighted network. The
traditional definition of network density is the ratio between the number of
existing links and the number of possible links (i.e. $N(N-1)/2$). However, in
a temporal hypernetwork, the subhypernetworks may not be connected, making the
dynamics solely occur in each component.

Here, we introduce the concept of connected density of networks to better
capture the internal structure. For $i$th component with $N_{i}$ nodes and
$L_{i}$ links, the density of the component is

$\rho_{i}=\frac{2L_{i}}{N_{i}(N_{i}-1)},$ (3)

which spans from 0 to 1. $\rho_{i}$ of an isolated node $i$ is defined as 0.
Then, the connected density of this subnetwork is defined as the corresponding
average over each component, namely,

$\rho=\sum_{i}\frac{N_{i}}{N}\rho_{i}.$ (4)

For the temporal hypernetwork or traditional temporal network, the average
density is $\langle\rho\rangle$ defined over each subhypernetwork. As Fig. 2c
shows, the purpose of this definition is to focus only on the connectivity
within components, which is the extent to which the dynamics can ripple. And
networks that are extremely sparse under the traditional definition can still
be dense from the perspective of their connected density.

We now explain our results by the connected density and hyperlink size induced
by $\tau$ and $\Delta t$. As the time interval $\tau$ of the subhypernetwork
gets longer, the average density of the temporal hypernetwork decreases
rapidly and then gradually increases. The connected density reaches the
minimum at a moderate $\tau$, and the results are displayed in Fig. 2d. That
is, for a moderate $\tau$, the hypernetwork is sparser, which can help
cooperators to aggregate together. Such a phenomenon is actually a phase
change process. If $\tau$ is extremely small, an increase in $\tau$
significantly increases the size of the components compared to the degree of
connectivity between nodes within the components. However, once the entire
network is close to fully connected, keeping increasing $\tau$ will only
enlarge the number of links, thus making the connected density larger.
Meanwhile, we find that increasing $\Delta t$ leads to an increase in
hyperlink size (Fig. 2e). And Eq. (2) shows that increasing the size of
hyperlinks will reduce the critical value $r_{C}$, which is consistent with
previous study [22].

To verify the generality of our conclusions, we perform additional simulations
on synthetic temporal hypernetworks with $p=0.05$ (Disconnected) and $0.6$
(Connected). The temporal hypernetwork with $p=0.05$ is disconnected for
almost all individuals and has similar properties and performance to the
empirical hypernetwork. The cooperation is promoted at a moderate $\tau$
(i.e., sparser hypernetworks), as the connected density decreases rapidly and
then turns to rise (Figs. 2f, g). The temporal hypernetwork of $p=0.6$
verifies our conclusions from another side, in which snapshots are all fully
connected. As $\tau$ goes up, the subhypernetworks become denser (Fig. 2j) and
inhibits cooperation (Fig. 2i). However, the size of hyperlinks is larger for
$p=0.6$ (Figs. 2f, j), making it easier for cooperation to emerge on temporal
hypernetworks (Figs. 2i, k).

Figure 3: Temporal hypernetworks can promote cooperation compared to their
static counterparts. We plot the frequency of cooperation $f_{C}$ as a
function of synergy factor $r$ on empirical (a) and synthetic (c, d)
hypernetworks. For synthetic hypernetwork, we set the probability $p$ for
links to be activated to be $0.6$ (Connected, c) and $0.05$ (Disconnected, d).
The frequency of cooperators is promoted as the round of interaction $g$
increases, and finally, exceeds the static counterparts. b. We present the
critical value of the synergy factor as a function of $g$. The magnitude of
the promotion of cooperation (i.e. reduction of $r_{C}$) is decreasing. For
pannel a, $\Delta t$=0.1h, $\tau=1$h. And $\tau=1$ for pannel c, $\tau=10$ for
pannel d. Other parameters are consistent with those used in Fig. 2.

###  III.2 Temporal hypernetworks can promote cooperation

Compared to static hypernetwork, temporal hypernetwork better models empirical
interactions by involving the interaction time. It is natural to ask what
impact will temporality have on the evolution of cooperation.

According to above results, for proper time interval $\tau$ the connected
density is lower than the static hypernetwork, which provides an opportunity
for the temporal hypernetwork to promote cooperation. The round of interaction
$g$ over each subhypernetwork stands for the similarity $\delta$ of the
temporal hypernetworks. If $g$ gets larger, the similarity is higher for not
changing subhypernetworks for a long time steps. Thus, from Eq. (2), we know
that increasing $g$ will enhance cooperation. Numerical results confirm this
analysis (Fig. 3a), and we find that, for a series of temporal hypernetworks
generated by proper $\Delta t$ and $\tau$ (sparser subhypernetworks), if $g$
is large, temporal hypernetworks facilitate cooperation more than their static
counterparts. As Fig. 3a shows, for $\Delta t=0.1$h, $\tau=1$h, such a
transition will occur between $g=100$ and $g=5000$. We further show that the
magnitude of the promotion is decreasing with the increase of $g$ (Fig. 3b).
As $g$ increases, cooperation is greatly facilitated, and the critical value
$r_{C}$ drops sharply. But when $g$ continues to increase, the decrease in the
critical values slows down. This can be explained by the fact that as $g$ goes
up, the rise of similarity $\delta$ of the temporal hypernetworks gradually
slows down (proportional to $1/g$). The same situation is observed on the
synthetic temporal hypernetworks (Fig. 3c, d). When $g=100$, all temporal
hypernetworks outperform their static counterparts. But by further increasing
$g$ from $100$ to $1000$, the promotion is negligible.

Intuitively, once the structure is defined, a fast evolutionary process, or,
in other words, having a long and fixed period of interaction (i.e. larger
$g$) helps the spread of cooperation. Such a phenomenon is directly linked to
the promotion of aggregations. When the interaction time $g$ is longer, it
allows for the aggregation of cooperators to be fully established. On the
other hand, rapid changes of subhypernetworks (i.e. smaller $g$) can disrupt
newly formed aggregations, thus discouraging cooperation.

Figure 4: The advantage of higher-order interactions on temporal and static
hypernetworks on favoring cooperation. We plot the frequency of cooperation
$f_{C}$ as a function of synergy factor $r$ on empirical hypernetwork and on
corresponding traditional network with pairwise interactions for $g=100$ (a)
and $g=5000$ (b). The $r_{C}$ is always smaller for temporal and static
hypernetworks than that for traditional networks. The other parameters are
consistent with those used in Fig. 2.

###  III.3 Advantage of higher-order network representation

After exploring the evolution of cooperation on the temporal hypernetworks, we
further consider advantage of higher-order network representation in promoting
cooperation. Specifically, we compare higher-order interactions on
hypernetworks with classical pairwise interactions on traditional networks.
For pairwise interactions, we construct the traditional temporal networks
under the same window $\Delta t=0.1$h and time interval $\tau=1$h through
empirical data sets, where the cliques are no longer considered as hyperlinks
[28]. The static network is undirected and unweighted, in which individuals
are linked once they interacted with each other. The game is the same as that
on hypernetworks, but the participants in each game are two individuals on
each link.

Our findings indicate that representing group interactions as a set of
pairwise interactions may lead to an underestimate of the potential for local
interactions to promote cooperation. We find that hypernetworks can
significantly facilitate cooperation compared to traditional networks based on
pairwise interactions for different settings of $g$ (Fig. 4). And cooperation
emerges with a smaller synergy factor $r_{C}$ in both temporal and static
hypernetworks. The reason may be that pairwise links on traditional networks
are smaller in size while at the same time, each individual has a larger
number of neighbors (namely, a larger degree) compared with hypernetworks.
Both factors may inhibit the evolution of cooperation as we have analyzed.

##  IV Discussion

By studying the evolution of cooperation on temporal hypernetworks, we
highlight the importance of temporal higher-order interactions on the fate of
cooperators. Since higher-order interactions can connect multiple individuals
at the same time, it is a reasonable way to represent group interactions. We
demonstrate that group interactions play a crucial role in facilitating
cooperation compared to traditional pairwise interactions. Also, higher-order
interactions embedded in hypernetworks are no longer restricted to the
previous assumptions on traditional networks, where the group could only be
defined over the focal individual and its first-order neighbors.

Indeed, group interactions are more complicated than pairwise ones [32]. In
microbial populations, the outcomes of 3-species cannot be fully predicted by
pairwise outcomes [33]. The complexity, such as the higher-order and time-
varying interactions, comes along with group interactions. Note that the
proposed framework of evolutionary dynamics on temporal hypernetworks not only
applies to public goods games but also to multi-strategy [34, 35, 36], multi-
player games with nonlinear and general payoff functions [37, 38, 39, 19, 40,
41], which are important research topics in the future.

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# Answers to questions of Grünbaum and Loewner

Sergii Myroshnychenko, Kateryna Tatarko, and Vladyslav Yaskin  S.
Myroshnychenko, Department of Mathematics and Statistics, University of the
Fraser Valley, Abbotsford, BC V2S 7M8, Canada<EMAIL_ADDRESS>K.
Tatarko, Department of Pure Mathematics, University of Waterloo, Waterloo, ON
N2L 3G1, Canada<EMAIL_ADDRESS>V. Yaskin, Department of Mathematical
and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
<EMAIL_ADDRESS>
###### Abstract.

We construct a convex body $K$ in $\mathbb{R}^{n}$, $n\geq 5$, with the
property that there is exactly one hyperplane $H$ passing through $c(K)$, the
centroid of $K$, such that the centroid of $K\cap H$ coincides with $c(K)$.
This provides answers to questions of Grünbaum and Loewner for $n\geq 5$. The
proof is based on the existence of non-intersection bodies in these
dimensions.

###### Key words and phrases:

Convex body, section, centroid, Fourier transform, non-intersection body

###### 2020 Mathematics Subject Classification:

Primary 52A20, 52A40

The authors were supported in part by NSERC

##  1\. Introduction

Let $K$ be a convex body in $\mathbb{R}^{n}$, i.e., a compact convex set with
non-empty interior. The centroid of $K$ (also known as the center of mass or
barycenter) is the point

$c(K)=\frac{1}{|K|}\int_{K}x\,dx,$

where integration is with respect to Lebesgue measure and $|K|$ denotes the
volume of $K$.

Consider the subset of the interior of $K$ with the property that each point
$p$ from this set is the centroid of at least $n+1$ different
$(n-1)$-dimensional sections of $K$ through $p$. Grünbaum [Gr3, §6.2] has
shown that this set is non-empty for every convex body $K$ and asked whether
the centroid $c(K)$ of $K$ belongs to this set; see [Gr2] and [CFG, A8]. For
the history of the problem and related results we refer the reader to [Gr3].

######  Problem 1.

(Grünbaum) Is the centroid $c(K)$ of $K\subset\mathbb{R}^{n}$ the centroid of
at least $n+1$ different $(n-1)$-dimensional sections of $K$ through $c(K)$?

A more general question was asked by Loewner [F, Problem 28] a few years
later.

######  Problem 2.

(Loewner) Let $\mu(K)$ be the number of hyperplane sections of $K$ passing
through $c(K)$ whose centroid is the same as $c(K)$. Let

$\mu(n)=\displaystyle\min_{K\in\mathcal{K}^{n}}\mu(K),$

where $\mathcal{K}^{n}$ is the class of all convex bodies in $\mathbb{R}^{n}$.
What is the value of $\mu(n)$?

It is a known fact that Problem 1 has a positive answer in $\mathbb{R}^{2}$.
We include a proof in Section 3 for completeness. Grünbaum [Gr2] has shown
that every point in the interior of a convex body is the centroid of at least
one section. Thus $\mu(n)\geq 1$ for all $n\geq 3$. In this paper we show that
$\mu(n)=1$ for $n\geq 5$. In particular, Problem 1 has a negative answer for
$n\geq 5$. We construct a convex body of revolution $K$ satisfying $\mu(K)=1$
using Fourier analytic tools and exploiting the fact that there are origin-
symmetric convex bodies that are non-intersection bodies in $\mathbb{R}^{n}$
for $n\geq 5$.

##  2\. Preliminaries

We say that a compact set $K\subset\mathbb{R}^{n}$ is star-shaped about the
origin $0$ if for every point $x\in K$ each point of the interval $[0,x)$ is
an interior point of $K$. The Minkowski functional of $K$ is defined by

$\|x\|_{K}=\min\\{\lambda\geq 0:x\in\lambda K\\}.$

$K$ is called a star body if it is compact, star-shaped about the origin and
its Minkowski functional is a continuous function on $\mathbb{R}^{n}$.

The radial function of a star body $K\subset\mathbb{R}^{n}$ is defined by

$\rho_{K}(\xi)=\max\\{\lambda>0:\lambda\xi\in K\\},\quad\xi\in S^{n-1}.$

It is clear that $\rho_{K}(\xi)=\|\xi\|_{K}^{-1}$ for any $\xi$ on the unit
sphere $S^{n-1}$ and $\rho_{K}$ is positive and continuous on $S^{n-1}$.

We say that $K$ is origin-symmetric if $x\in K$ if and only if $-x\in K$. For
an origin-symmetric star body $K$ we have $\rho_{K}(\xi)=\rho_{K}(-\xi)$ for
all $\xi\in S^{n-1}$.

One of the main tools in this paper is the Fourier transform of distributions.
The reader is referred to [GS] and [K2] for background information. Let
$\mathcal{S}(\mathbb{R}^{n})$ be the Schwartz space of rapidly decreasing
infinitely differentiable functions on $\mathbb{R}^{n}$, called test
functions. By $\mathcal{S}^{\prime}(\mathbb{R}^{n})$ we denote the space of
distributions, i.e., continuous linear functionals on
$\mathcal{S}(\mathbb{R}^{n})$. For a test function
$\varphi\in\mathcal{S}(\mathbb{R}^{n})$, its Fourier transform is given by

$\widehat{\varphi}(x)=\int_{\mathbb{R}^{n}}\varphi(y)e^{-i\langle
x,y\rangle}\,dy.$

For a distribution $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ its Fourier
transform $\widehat{f}$ (also denoted by $f^{\wedge}$) is a distribution
defined by

$\langle\widehat{f},\varphi\rangle=\langle{f},\widehat{\varphi}\rangle,$

for every $\varphi\in\mathcal{S}(\mathbb{R}^{n})$. Note that for any even
distribution $f$ we have $\widehat{\widehat{f}\,\,}=(2\pi)^{n}f$.

If $f$ is a homogeneous function of degree $-k$ on
$\mathbb{R}^{n}\setminus\\{0\\}$, where $0<k<n$, then $f$ can be thought of as
a distribution acting on test functions by integration. In addition, if $f\in
C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$, then $\widehat{f}$ is a
homogeneous function of degree $-n+k$ which is infinitely differentiable on
$\mathbb{R}^{n}\setminus\\{0\\}$; see [K2, Lemma 3.6]. If $f$ is an even
homogeneous function of degree $-n+1$, then its Fourier transform restricted
to the sphere can be computed as follows (see [K2, Lemma 3.7]):

$\widehat{f}(\xi)=\pi\int_{S^{n-1}\cap\xi^{\perp}}f(x)\,dx,\qquad\xi\in
S^{n-1},$ (1)

where $\xi^{\perp}=\\{x\in\mathbb{R}^{n}:\langle x,\xi\rangle=0\\}$ is the
hyperplane through the origin orthogonal to the vector $\xi$, and
$\langle\cdot,\cdot\rangle$ stands for the standard inner product on
$\mathbb{R}^{n}$.

We will denote the Euclidean norm of a vector $x\in\mathbb{R}^{n}$ by $|x|$.
The Fourier transform of $|x|^{-1}$ equals (see [GS, p. 363])

$\left(|\cdot|^{-1}\right)^{\wedge}(x)=c_{n}|x|^{-n+1},$ (2)

where

$c_{n}={2^{n-1}\pi^{\frac{n-1}{2}}\Gamma\left(\frac{n-1}{2}\right)}.$ (3)

Note that if $T$ is an invertible linear transformation on $\mathbb{R}^{n}$,
then

$(|Tx|^{-1})^{\wedge}(y)=c_{n}|\det T|^{-1}|T^{-t}y|^{-n+1},$ (4)

where $T^{-t}$ denotes the inverse of the transpose of $T$.

Let $f$ and $g$ be infinitely differentiable even functions on
$\mathbb{R}^{n}\setminus\\{0\\}$. Assume that $f$ and $g$ are homogeneous of
degrees $-k$ and $-n+k$, respectively, for some $0<k<n$. Then the following
spherical Parseval formula holds:

$\int_{S^{n-1}}\widehat{f}(\xi)g(\xi)\,d\xi=\int_{S^{n-1}}f(\xi)\widehat{g}(\xi)\,d\xi.$
(5)

This formula is proven in [K2, Lemma 3.22] in the case when
$f=\|\cdot\|_{K}^{-k}$ and $g=\|\cdot\|_{L}^{-n+k}$ for some star bodies $K$
and $L$, but this immediately yields the case of general functions.
Alternatively, one can prove (5) using spherical harmonics; see [GYY] for
details.

The notion of the intersection body of a star body was first introduced by
Lutwak [L] and has played an important role in convex geometry. Let $K$ and
$L$ be origin-symmetric star bodies in $\mathbb{R}^{n}$. We say that $K$ is
the intersection body of $L$ if

$\rho_{K}(\xi)=|L\cap\xi^{\perp}|,\quad\mbox{for all }\xi\in S^{n-1}.$

Passing to polar coordinates, we see that

$\rho_{K}(\xi)=\frac{1}{n-1}\int_{S^{n-1}\cap\xi^{\perp}}\rho_{L}^{n-1}(x)\,dx.$

By formula (1) this means that

$\|\xi\|_{K}^{-1}=\frac{1}{\pi(n-1)}\left(\|\cdot\|_{L}^{-n+1}\right)^{\wedge}(\xi),$

which implies that

$\left(\|\cdot\|_{K}^{-1}\right)^{\wedge}(\xi)=\frac{(2\pi)^{n}}{\pi(n-1)}\
{\|\xi\|_{L}^{-n+1}}>0.$

More generally, $K$ is called an intersection body if $\rho_{K}$ is the
spherical Radon transform of a positive measure on $S^{n-1}$; see [GLW].
Alternatively, the class of intersection bodies can be defined as the closure
of the class of intersection bodies of star bodies in the radial metric.
Intersection bodies were a key ingredient in the solution of the celebrated
Busemann–Petty problem; the connection was discovered by Lutwak [L]. The
following Fourier analytic characterization of intersection bodies was
obtained by Koldobsky [K1]. An origin-symmetric star body in $\mathbb{R}^{n}$
is an intersection body if and only if $\|\cdot\|_{K}^{-1}$ is a positive
definite distribution, i.e., the Fourier transform of $\|\cdot\|_{K}^{-1}$ is
a positive distribution. All origin-symmetric convex bodies in dimensions 2,
3, and 4 are intersection bodies, while in dimensions $n\geq 5$ there are non-
intersection bodies. Together with the Lutwak connection this gives a positive
answer to the Busemann-Petty problem in dimensions $n\leq 4$ and a negative
answer in dimensions $n\geq 5$ (see [K2] for the history of the problem and a
Fourier analytic solution). For various constructions of non-intersection
bodies see [Ga], [Z], [K1], [GKS], [Ya], [Y].

##  3\. The planar case

For $n=2$, Problem 1 has an affirmative answer. In particular, the centroid of
any triangle bisects exactly those chords that are parallel to the sides. In
general, the centroid $c(K)$ of any planar convex body
$K\subset\mathbb{R}^{2}$ bisects at least three different chords of $K$. This
is well known, but we provide a proof for completeness. Without loss of
generality, we can choose the centroid of $K$ to be the origin, that is,

$\int_{0}^{2\pi}\rho^{3}(\theta)\cos(\theta)d\theta=0\quad\text{and}\quad\int_{0}^{2\pi}\rho^{3}(\theta)\sin(\theta)d\theta=0,$

where the radial function $\rho$ of $K$ is given as a function of the polar
angle $\theta$. Equivalently,

$\int_{0}^{\pi}f(\theta)\cos(\theta)d\theta=0\quad\text{and}\quad\int_{0}^{\pi}f(\theta)\sin(\theta)d\theta=0,$
(6)

where $f(\theta)=\rho^{3}(\theta)-\rho^{3}(\theta+\pi)$ is a continuous
function on $\mathbb{R}$ with period $2\pi$. Since $f(\theta)=-f(\theta+\pi)$,
there is at least one root $\theta_{0}\in[0,\pi)$ where $f$ changes its sign.
If such a root were unique then the function
$f(\theta)\sin(\theta-\theta_{0})$ would not change its sign in the interval
$[0,\pi]$. However, this is impossible since
$\int_{0}^{\pi}f(\theta)\sin(\theta-\theta_{0})d\theta=0$ by (6). Thus, $f$
has at least two distinct roots $\theta_{0}$ and $\theta_{1}$
($\theta_{0}<\theta_{1}$) on $[0,\pi)$, where $f$ changes its sign. Note that
$f$ cannot have exactly two such roots. If this were the case, $f$ would have
the same sign on the intervals $(\theta_{1}-\pi,\theta_{0})$ and
$(\theta_{1},\theta_{0}+\pi)$ which is impossible since
$f(\theta)=-f(\theta+\pi)$ for all $\theta$. Thus,
$\rho(\theta)=\rho(\theta+\pi)$ for at least three distinct values in the
interval $[0,\pi)$ which implies that $c(K)$ splits at least three chords in
the corresponding directions in half.

##  4\. Main result

######  Theorem 3.

There exists a convex body $K\subset\mathbb{R}^{n}$, $n\geq 5$, with centroid
at the origin, such that

$\langle c(K\cap\xi^{\perp}),e_{n}\rangle\geq 0\quad\text{for any}\quad\xi\in
S^{n-1},$

with equality if and only if $\xi=\pm e_{n}$, where $e_{n}=(0,\ldots,0,1)$.

We begin with an auxiliary construction. For a small $a>0$, we define an
origin-symmetric non-intersection convex body of revolution $M$ by

$\|x\|^{-1}_{M}=|x|^{-1}-2a^{n-2}\left(x_{1}^{2}+\cdots+x_{n-1}^{2}+\frac{x_{n}^{2}}{a^{2}}\right)^{-\frac{1}{2}},\quad\text{for
}x\in\mathbb{R}^{n}\setminus\\{0\\}.$

This is a particular case of the construction from [Y, Theorem 2.2]. We note
that
$|x|^{-1}>\left(x_{1}^{2}+\cdots+x_{n-1}^{2}+\frac{x_{n}^{2}}{a^{2}}\right)^{-\frac{1}{2}}$
for small $a>0$ and all $x\in\mathbb{R}^{n}\setminus\\{0\\}$, so
$\|x\|^{-1}_{M}>0$ on $\mathbb{R}^{n}\setminus\\{0\\}$ and $M$ is well-
defined.

######  Lemma 4.

$M$ is a convex body with strictly positive curvature when $a$ is sufficiently
small.

###### Proof.

This is a standard perturbation argument; cf. [K2, p. 96]. To show that $M$
has strictly positive curvature, observe that $M$ is a small perturbation of
the Euclidean ball whose curvature is 1 everywhere. We can control the first
and second derivatives of the perturbation function
$2a^{n-2}\left(x_{1}^{2}+\cdots+x_{n-1}^{2}+\frac{x_{n}^{2}}{a^{2}}\right)^{-\frac{1}{2}}$.
These derivatives are of order $a^{n-4}$ which are small for sufficiently
small $a>0$ (since $n\geq 5$). Hence, $M$ has strictly positive curvature. ∎

From now on we will fix a value of $a$ for which Lemma 4 is true.

######  Lemma 5.

$M$ is not an intersection body. In particular,
$\left(\|\cdot\|^{-1}_{M}\right)^{\wedge}(\pm e_{n})<0$.

###### Proof.

To compute the Fourier transform of $\|x\|^{-1}_{M}$ we use formulas (2) and
(4):

$\left(\|\cdot\|^{-1}_{M}\right)^{\wedge}(x)=c_{n}\left(|x|^{-n+1}-2a^{n-1}\left(x_{1}^{2}+\cdots+x_{n-1}^{2}+a^{2}x_{n}^{2}\right)^{\frac{-n+1}{2}}\right),$

where $c_{n}$ is the positive constant defined in (3). It remains to observe
that

$\left(\|\cdot\|^{-1}_{M}\right)^{\wedge}(\pm e_{n})=-c_{n}<0.$

∎

Proof of Theorem 3. By Lemma 5,
$\left(\|\cdot\|^{-1}_{M}\right)^{\wedge}(e_{n})<0$. Since
$\left(\|\cdot\|^{-1}_{M}\right)^{\wedge}$ is continuous on the sphere
$S^{n-1}$, there is an open spherical ball $\Omega(e_{n})\subset S^{n-1}$
centered at $e_{n}$ such that
$\left(\|\cdot\|^{-1}_{M}\right)^{\wedge}(\xi)<0$ for any
$\xi\in\Omega(e_{n}).$ Define $\Omega(-e_{n})$ to be the reflection of
$\Omega(e_{n})$ with respect to the center of the sphere. Let $G\in
C^{\infty}(S^{n-1})$ be an even function that is invariant under rotations
about the $x_{n}$-axis and that satisfies the following properties:

$G(\xi)=\begin{cases}\text{positive},\quad\xi\in\Omega(\pm
e_{n})\backslash\\{\pm e_{n}\\};\\\ 0,\quad\xi\in\\{\pm e_{n}\\}\cup
S^{n-1}\backslash\Omega(\pm e_{n}).\end{cases}$

By construction,

$\int_{S^{n-1}}\left(\|\cdot\|^{-1}_{M}\right)^{\wedge}(\xi)G(\xi)d\xi<0.$ (7)

Next, we define the following function $H\in C^{\infty}(S^{n-1})$ that is
invariant under rotations about the $x_{n}$-axis:

$H(x)=|x|^{-1}-(4(x_{1}^{2}+\dots+x_{n-1}^{2})+x_{n}^{2})^{-\frac{1}{2}}.$

Note that $H(x)>0$ when $x\in S^{n-1}\setminus\\{\pm e_{n}\\}$, and $H(\pm
e_{n})=0$. Extending $H$ to $\mathbb{R}^{n}\backslash\\{0\\}$ as a homogeneous
function of degree $-1$ and using formulas (2) and (4), we obtain

$\displaystyle\widehat{H}(x)$
$\displaystyle=c_{n}\left(|x|^{-n+1}-\frac{1}{2^{n-1}}\left(\frac{x_{1}^{2}+\dots+x_{n-1}^{2}}{4}+x_{n}^{2}\right)^{\frac{-n+1}{2}}\right)$
$\displaystyle=c_{n}\left(|x|^{-n+1}-\left(x_{1}^{2}+\dots+x_{n-1}^{2}+4x_{n}^{2}\right)^{\frac{-n+1}{2}}\right).$

Observe that $\widehat{H}(x)\geq 0$ for $x\in\mathbb{R}^{n}\setminus\\{0\\}$.
Hence, using the spherical Parseval formula (5), we get

$\int_{S^{n-1}}\left(\|\cdot\|^{-1}_{M}\right)^{\wedge}(\xi)H(\xi)\,d\xi=\int_{S^{n-1}}\|\xi\|^{-1}_{M}\widehat{H}(\xi)\,d\xi>0.$
(8)

Finally, for $\lambda\in[0,1]$ we define

$g_{\lambda}(\xi)=(1-\lambda)G(\xi)+\lambda H(\xi),\qquad\xi\in S^{n-1}.$

Since $G(\xi)\geq 0$ and $H(\xi)>0$ for all $\xi\in S^{n-1}\setminus\\{\pm
e_{n}\\}$, and $G(\pm e_{n})=H(\pm e_{n})=0$, we see that for all
$\lambda\in(0,1]$ the function $g_{\lambda}$ satisfies the properties:
$g_{\lambda}(\xi)>0$ for all $\xi\in S^{n-1}\setminus\\{\pm e_{n}\\}$, and
$g_{\lambda}(\pm e_{n})=0$.

Since $G$ and $H$ are $C^{\infty}$ functions on the sphere, so is
$g_{\lambda}$. Extending $g_{\lambda}$ to $\mathbb{R}^{n}\setminus\\{0\\}$ as
a homogeneous function of degree $-1$ and denoting the Fourier transform of
this extension by $\widehat{g_{\lambda}}$, we define a function
$\phi_{\lambda}$ on $S^{n-1}$ by the formula

$\phi_{\lambda}(\xi)=\frac{1}{\xi_{n}}\widehat{g_{\lambda}}(\xi),\qquad\xi\in
S^{n-1},$

where $\xi_{n}=\langle\xi,e_{n}\rangle$. Note that $\phi_{\lambda}$ does not
have a singularity when $\xi_{n}=0$, since $\widehat{g_{\lambda}}(\xi)=0$ for
all $\xi\in e_{n}^{\perp}$. To show the latter, observe that
$\widehat{g_{\lambda}}$ is rotationally invariant as $g_{\lambda}$ has the
same property. Therefore for every $\xi\in e_{n}^{\perp}$ we have

$\displaystyle\widehat{g_{\lambda}}(\xi)$
$\displaystyle=\frac{1}{|S^{n-2}|}\int_{S^{n-1}\cap
e_{n}^{\perp}}\widehat{g_{\lambda}}(\xi)\,dx=\frac{1}{|S^{n-2}|}\int_{S^{n-1}\cap
e_{n}^{\perp}}\widehat{g_{\lambda}}(x)\,dx$
$\displaystyle=\frac{1}{\pi|S^{n-2}|}\,\widehat{\widehat{g_{\lambda}}}(e_{n})=\frac{(2\pi)^{n}}{\pi|S^{n-2}|}\,g_{\lambda}(e_{n})=0,$

where we used formula (1).

Since $\widehat{g_{\lambda}}$ is even, infinitely smooth, and vanishes on
$e_{n}^{\perp}$, we can define $\phi_{\lambda}$ to be zero on $e_{n}^{\perp}$.
The function $\phi_{\lambda}$ constructed this way is odd and infinitely
smooth. Also observe that the first and second derivatives of
$\widehat{g_{\lambda}}$ are bounded on the sphere, uniformly in
$\mathbb{\lambda}\in[0,1]$. This is due to the fact that $\phi_{\lambda}$ is
linear in $\lambda$.

For a small $\varepsilon>0$ define a body $K$ as follows:

$\rho_{K}(\xi)=\left(\rho_{M}^{n}(\xi)+\varepsilon\phi_{\lambda}(\xi)\right)^{1/n},\qquad\xi\in
S^{n-1}.$

Since $M$ has strictly positive curvature and the derivatives of
$\phi_{\lambda}$ are bounded on the sphere, uniformly in $\lambda$, we
conclude that there is a small $\alpha>0$ such that the body $K$ is convex for
all $\varepsilon\in[0,\alpha]$ and all $\lambda\in[0,1]$. Additionally, $K$ is
a body of revolution about the $x_{n}$-axis.

Now observe that the centroids of sections $K\cap\xi^{\perp}$ all lie in the
open half-space $\\{x\in\mathbb{R}^{n}:\,x_{n}>0\\}$, unless $\xi=\pm e_{n}$.
Indeed, for all $\xi\in S^{n-1}\setminus\\{\pm e_{n}\\}$ we have

$\displaystyle|K\cap\xi^{\perp}|\,\langle c(K\cap\xi^{\perp}),e_{n}\rangle$
$\displaystyle=\int_{K\cap\xi^{\perp}}x_{n}\,dx$
$\displaystyle=\frac{1}{n}\int_{S^{n-1}\cap\xi^{\perp}}x_{n}\rho_{K}^{n}(x)\,dx$
$\displaystyle=\frac{1}{n}\int_{S^{n-1}\cap\xi^{\perp}}x_{n}\left(\rho_{M}^{n}(x)+\varepsilon\phi_{\lambda}(x)\right)\,dx$
$\displaystyle=\frac{1}{n}\int_{S^{n-1}\cap\xi^{\perp}}x_{n}\varepsilon\phi_{\lambda}(x)\,dx$
$\displaystyle=\frac{\varepsilon}{\pi
n}\left(x_{n}\phi_{\lambda}(x)\right)^{\wedge}(\xi)=\frac{\varepsilon(2\pi)^{n}}{\pi
n}g_{\lambda}(\xi)>0,$

where we used the origin-symmetry of $M$ and formula (1). Note that the latter
strict inequality holds for all $\varepsilon>0$ and all $\lambda\in(0,1]$.

Now we will choose $\lambda$ and $\varepsilon$ so that the centroid of $K$ is
at the origin. Since $K$ is a body of revolution, it is enough to ensure that
the quantity below can be equal to zero.

$\displaystyle|K|\,\langle c(K),e_{n}\rangle$
$\displaystyle=\int_{K}x_{n}\,dx=\frac{1}{n+1}\int_{S^{n-1}}x_{n}\rho_{K}^{n+1}(x)\,dx$
$\displaystyle=\frac{1}{2(n+1)}\int_{S^{n-1}}x_{n}\left(\rho_{K}^{n+1}(x)-\rho_{K}^{n+1}(-x)\right)\,dx$
$\displaystyle=\frac{1}{2(n+1)}\int_{S^{n-1}}\left(\frac{\rho_{K}^{n+1}(x)-\rho_{K}^{n+1}(-x)}{\varepsilon\phi_{\lambda}(x)}\right)x_{n}\varepsilon\phi_{\lambda}(x)\,dx.$

Since $\phi_{\lambda}$ is odd and $M$ is origin-symmetric, we have

$\displaystyle\frac{\rho_{K}^{n+1}(x)-\rho_{K}^{n+1}(-x)}{\varepsilon\phi_{\lambda}(x)}$
$\displaystyle=\frac{\left(\rho_{M}^{n}(x)+\varepsilon\phi_{\lambda}(x)\right)^{\frac{n+1}{n}}-\left(\rho_{M}^{n}(-x)+\varepsilon\phi_{\lambda}(-x)\right)^{\frac{n+1}{n}}}{\varepsilon\phi_{\lambda}(x)}$
$\displaystyle=\frac{\left(\rho_{M}^{n}(x)+\varepsilon\phi_{\lambda}(x)\right)^{\frac{n+1}{n}}-\left(\rho_{M}^{n}(x)-\varepsilon\phi_{\lambda}(x)\right)^{\frac{n+1}{n}}}{\varepsilon\phi_{\lambda}(x)}$
$\displaystyle=\rho_{M}^{n+1}(x)\frac{\left(1+\varepsilon\phi_{\lambda}(x)\rho_{M}^{-n}(x)\right)^{\frac{n+1}{n}}-\left(1-\varepsilon\phi_{\lambda}(x)\rho_{M}^{-n}(x)\right)^{\frac{n+1}{n}}}{\varepsilon\phi_{\lambda}(x)}$
$\displaystyle=\rho_{M}^{n+1}(x)\frac{\frac{2(n+1)}{n}\varepsilon\phi_{\lambda}(x)\rho_{M}^{-n}(x)+\varepsilon^{3}\phi_{\lambda}^{3}(x)R_{1}(x)}{\varepsilon\phi_{\lambda}(x)}$
$\displaystyle=\frac{2(n+1)}{n}\left(\rho_{M}(x)+\varepsilon^{2}R(x)\right),$

where $R_{1}$ is the remainder in the Taylor expansion in $\varepsilon$ of the
corresponding expression above, and
$R(x)=\frac{n}{2(n+1)}\rho_{M}^{n+1}(x)\phi_{\lambda}^{2}(x)R_{1}(x)$.

Thus,

$\displaystyle|K|\,\langle c(K),e_{n}\rangle$
$\displaystyle=\frac{1}{n}\int_{S^{n-1}}\left(\rho_{M}(x)+\varepsilon^{2}R(x)\right)x_{n}\varepsilon\phi_{\lambda}(x)\,dx$
$\displaystyle=\frac{\varepsilon}{n}\int_{S^{n-1}}\left(\rho_{M}(x)+\varepsilon^{2}R(x)\right)\widehat{g_{\lambda}}(x)\,dx$
$\displaystyle=\frac{\varepsilon}{n}\int_{S^{n-1}}\left(\widehat{\rho_{M}}(\xi)+\varepsilon^{2}\widehat{R}(\xi)\right){g_{\lambda}}(\xi)\,d\xi,$

where we used the spherical Parseval formula (5).

Define

$F(\lambda,\varepsilon)=\int_{S^{n-1}}\left(\widehat{\rho_{M}}(\xi)+\varepsilon^{2}\widehat{R}(\xi)\right){g_{\lambda}}(\xi)\,d\xi.$

Note that all derivatives of $R$ depend continuously on $\lambda$ and
$\varepsilon$. Therefore, by [K2, Lemma 3.16], $\widehat{R}$ is also a
continuous function of $\lambda$ and $\varepsilon$, and hence so is $F$ for
$(\lambda,\varepsilon)\in[0,1]\times[0,\alpha].$

Observe that

$F(\lambda,0)=\int_{S^{n-1}}\widehat{\rho_{M}}(\xi)g_{\lambda}(\xi)\,d\xi$

is a sign-changing function of $\lambda\in[0,1]$. Indeed,

$F(0,0)=\int_{S^{n-1}}\widehat{\rho_{M}}(\xi)G(\xi)\,d\xi<0$

and

$F(1,0)=\int_{S^{n-1}}\widehat{\rho_{M}}(\xi)H(\xi)\,d\xi>0$

by (7) and (8) respectively.

Since $F(\lambda,\varepsilon)$ is a continuous function on
$[0,1]\times[0,\alpha]$, there exist a small $\varepsilon_{0}>0$ and
$\lambda_{0}\in(0,1)$ such that $F(\lambda_{0},\varepsilon_{0})=0$, as
required.

∎

######  Remark 1.

The above construction does not work when $n=3$ and $n=4$ since there are no
non-intersection bodies in these dimensions. We suspect that
$\mu(3)=\mu(4)=1$, but the construction has to be more delicate.

######  Remark 2.

This work continues a recent trend of studying relative positions of various
centroids; see e.g., [MTY] and [NRY]. In [NRY] the authors found an optimal
upper bound on the distance between the centroid of a planar body and its
boundary. The corresponding question is open in $\mathbb{R}^{n}$, $n\geq 3$.
In [MTY] the authors found an optimal upper bound on the distance between a
projection of the centroid of a convex body $K$ and the centroid of the
projection of $K$. An analogue of the latter question for sections is stated
in [S] and is currently open. One of the main motivations for such questions
comes from Grünbaum-type inequalities; see [Gr1], [FMY], [SZ], [MNRY], [MSZ].

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