Schedule for: 23w5105 - Interplay between Geometric Analysis and Discrete Geometry

The Kneser-Poulsen conjecture was originally proposed by Kneser (1955) and Poulsen (1954) independently as a fundamental geometric problem on finite systems of sphere. On the other hand, ball-polyhedra are intersections of finitely many congruent balls. Both topics have been studied from the point of view of convex and discrete geometry. The talk aims to bridge them via a selection of old and new results.

Borsuk's number $f(n)$ is the smallest integer such that any set of diameter 1 in the $n$-dimensional Euclidean space can be covered by $f(n)$ sets of smaller diameter. Currently best known asymptotic upper bound $f(n)\le (\sqrt{3/2}+o(1))^n$ was obtained by Shramm (1988) and by Bourgain and Lindenstrauss (1989) using different approaches. Bourgain and Lindenstrauss estimated the minimal number $g(n)$ of open balls of diameter 1 needed to cover a set of diameter 1 and showed $1.0645^n\le g(n)\le (\sqrt{3/2}+o(1))^n$. On the other hand, Schramm used the connection $f(n)\le h(n)$, where $h(n)$ is the illumination number of $n$-dimensional convex bodies of constant width, and showed $h(n)\le (\sqrt{3/2}+o(1))^n$. The best known asymptotic lower bound on $h(n)$ is subexponential and is the same as for $f(n)$, namely $h(n)\ge f(n)\ge c^{\sqrt{n}}$ for large $n$ established by Kahn and Kalai with $c\approx 1.203$ (1993) and by Raigorodskii with $c\approx 1.2255$ (1999). In 2015 Kalai asked if an exponential lower bound on $h(n)$ can be proved. We show $h(n)\ge (\cos(\pi/14)+o(1))^{-n}$ by constructing the corresponding $n$-dimensional bodies of constant width, which answers Kalai's question in the affirmative. The construction is based on a geometric argument combined with a probabilistic lemma establishing the existence of a suitable covering of the unit sphere by equal spherical caps having sufficiently separated centers. The lemma also allows to improve the lower bound of Bourgain and Lindenstrauss to $g(n)\ge (2/\sqrt{3}+o(1))^{n} \approx 1.1547^n$. The talk is based on a joint work with Andrii Arman and Andriy Bondarenko. 

There have been several results recently concerning central limit theorems for random polytopes in various geometric settings that were proved via Stein's method. Some of the key ingredients of these arguments are floating bodies and their visibility regions. In the last few years, there has also been quite much work done on random polytope models in which the notion of convex hull was modified. The combination of these two topics raises questions about generalizations of floating bodies and their use in proving limit theorems and more. As an application we show a quantitative central limit theorem for the area of random disc-polygons, and raise several open questions. Part of this talk is joint work with Dániel Papvári (Szeged, Hungary).

Given a convex domain $C$, a $C$-polygon is an intersection of $n>1$ homothets of $C$. In this talk we explore the boundary structure of $C$-polygons and see how different properties on the boundary of $C$, such as smoothness or strict convexity, imply bounds on how complex the boundary of a $C$-polygon can be.

Different quantitative versions of classical convexity results have recently gained attention. In this talk, we will focus on the coarse approximation of polytopes. Specifically, we will show that the convex hull of a carefully selected set of $2d$ vertices from a well-centered polytope $P \subset \mathbb{R}^d$ contains a homothet $c(d)P$ of the original polytope $P$. The proof of this result requires only a basic understanding of linear algebra. However, the implications of this finding extend beyond its simplicity. It sheds light on quantitative Helly-type and Caratheodory-type results, The talk is based on two joint works with Márton Naszódi.

We call a convex body $K$ reduced, if for any different convex body contained in $K$ has a smaller minimal width. Reduced bodies are extremizers to some inequalities in convex geometry, and they also give a different perspective to the broadly studied family of bodies of constant width. There are multiple recent studies about reduced bodies in Minkowski spaces and spherical reduced bodies, and we also present a hyperbolic approach after introducing an extended version of Leichtweiss' width function. We prove the hyperbolic version of Pál's inequality, stating that the regular triangle has the smallest area among convex bodies of minimal width, and we will also discuss a stability version of the theorem. The talk is based on joint works with Károly J. Böröczky, András Csépai and Ansgar Freyer.

For a convex body $K$ in $\mathbb R^n$ its parallel section function is given by $A_{K,\xi} (t)= |K\cap (\xi^\perp + t\xi)|$, where $\xi \in S^{n-1}$ and $t\in \mathbb R$. We say that $K$ is polynomially integrable if $A_{K,\xi} (t)$ is a polynomial of $t$ on its support. It was shown by Koldobsky, Merkurjev, and Yaskin that the only polynomially integrable bodies are ellipsoids in odd dimensions. In even dimensions such bodies do not exist. In this talk we will discuss an analogue of polynomially integrable bodies in even dimensions: these are the bodies for which the Hilbert transform of $A_{K,\xi} (t)$ is a polynomial of $t$ (on an appropriate interval). We will see that ellipsoids in even dimensions are the only convex bodies satisfying this property. Joint work with M. Agranovsky, A. Koldobsky, and D. Ryabogin.

Convex polyhedron in $\mathbb{E}^d$ is a bounded intersection of a finite set of halfspaces. A polyhedron is facet-lean if omitting any halfspace from the intersecting set makes the intersection unbounded. In this talk I will classify the facet-lean polyhedra, prove that any facet-lean polyhedron in $\mathbb{E}^d$ can be illuminated by $2^d$ light sources, and will outline possible directions that might lead to proving the illumination conjecture for convex polyhedra in general case.

Given $n$ homothetic copies of a convex body with disjoint interiors, it is natural to ask how many possible contacts can occur between them. For example, Oded Schramm proved in his thesis and other early research that (with some special exceptions) every planar graph can be realised as the contact graph of such a (2-dimensional) packing. However, this requires that the scaling for each homothetic copy of the convex body is very carefully selected; for example, while a disc can be chosen to lie in the interior of three pairwise-touching discs, the radius of such a disc is essentially fixed. Connelly, Gortler and Theran proved in 2019 that if the radii of a disc packing are randomly selected, then the packing can have at most $2n-3$ contacts, a significant departure from the maximum achievable count of $3n-6$. In my talk I will discuss my own analogous results on the topic when dealing with a variety of types of convex bodies in 2-dimensions and higher, including: smooth and strictly convex centrally symmetric convex bodies, squares, convex bodies with positive curvature boundaries, spheres and cubes.

We introduce a function on order types that measures their rigidity with respect to projective transformations and give some examples. We show that for large $n$ the $n \times n$ grid is rigid, namely that every orientation preserving map of the $n \times n$ grid is $O(\frac1n )$ close to a projective transformation. Other examples include the order types on at most five points, some convex $n$-gons and the square grid on $9$ and $16$ points.

We discuss volume ratios between convex bodies and their projections. We recall known results and then show that for every $n$-dimensional convex body $K$ there exists a centrally-symmetric convex body $L$ such that for any two projections $P, Q$ of rank $k\leq n$ the volume ratio between $PK$ and $QL$ is large. Our result is sharp (up to logarithmic factors) when $k\geq n^{2/3}$. This is a joint work with D.Galicer, M. Merzbacher, and D. Pinasco.

A boundary point $q$ is an equilibrium point of a $3$-dimensional convex body $K$ with center of mass $c$ if the plane through $q$ and orthogonal to the segment $[q,c]$ supports $K$. In this case, in particular, $K$ can be balanced on a horizontal plane in a position touching it at $q$. Assuming nondegeneracy of the body, three types of equilibrium points are distinguished: stable, unstable and saddle-type. A consequence of the Poincar\'e-Hopf theorem is that the numbers $S$, $U$ and $H$ of the stable, unstable and saddle-type points of a $3$-dimensional convex body, respectively, satisfy the equation $S-H+U=2$. If a convex body has a unique stable or unstable point, it is called \emph{monostable} and \emph{mono-unstable}, respectively, and a body which is simultaneously monostable and mono-unstable is called \emph{mono-monostatic}. A famous example of a mono-monostatic convex body is the so-called \emph{G\"omb\"oc}, constructed by Domokos and V\'arkonyi. A 1969 question of Conway and Guy, also asked independently by Shephard in 1968, asks if there is a monostable convex polyhedron with a $k$-fold rotational symmetry, where $k > 2$. The answer to this problem was given by L\'angi in 2022, who proved the existence of such a polyhedron for all positive integers $k > 2$. In this talk we strengthen this result by characterizing the possible symmetry groups of monostable, mono-unstable and mono-monostatic convex polyhedra and convex bodies. The results presented in the talk are a joint work with G. Domokos and P. V\'arkonyi.

In this talk we introduce some connections between the \emph{The Borsuk partition problem } and \emph{The V\'azsonyi problem}, two attractive and famous problems in discrete and combinatorial geometry, both related on the diameter of a bounded set $S\subset \mathbb{R}^3$. Borsuk's problem asks whether every set $S\subset \mathbb{R}^d$ with finite diameter $diam(S)$ is the union of $d+1$ sets of diameter less than $diam(S)$. For $\mathbb{R}^2$ and $\mathbb{R}^3$ it is true, but for dimensions greater than 63 is false. V\'azsonyi's problem on the other hand asks for the maximum number of diameters over all the set of $n$ points in $\mathbb {R}^d$. In $\mathbb{R}^3$, the answer is $2n-2$ and the configurations attaining this number are already known. We shall present an equivalence between the critical sets with Borsuk number 4 in $\mathbb{R}^3$ and the minimal structures for the V\'azsonyi problem. This is a joint work with D\'eborah Oliveros and Jorge Ram\'­irez Alfons\'in.

Denote by $\mathcal{K}_0^n$ the class of closed convex sets $A\subseteq\mathbb{R}^n$ containing the origin $0\in A$, and recall that the polar duality (or polarity) is the map on $\mathcal{K}_0^n$ sending $A\in\mathcal{K}_0^n$ to its polar set $A^\circ$. It is well-known that polarity is an involution on $\mathcal{K}_0^n$ with a unique fixed point, and that it reverses inclusions In this talk, we exhibit a topological characterization of the polar duality and describe its relation with the so called Anderson's Conjecture. The later is an open problem regarding the characterization of all continuous involutions with a unique fixed point on the Hilbert cube $Q=\prod_{i=1}^{\infty}[-1,1]$. To this end, we shall show that $\mathcal{K}_0^n$, endowed with the Attouch-Wets metric, is homeomorphic with $Q$ and the polar duality is topologically conjugate with the standard involution $\sigma:Q\rightarrow Q$ given by $\sigma(x)=-x$. On the geometric side, we shall also prove that among all involutions on $\mathcal{K}_0^n$ that reverse inclusions those and only those with a unique fixed point are topologically conjugate with the polar duality. This is a joint work with Natalia Jonard-Pérez.

It was proved by Boroczky and Peyerimhoff that among simplices inscribed in a ball in spherical and hyperbolic space, respectively, the regular simplices have maximal volume. In my talk I will show our result that among simplices circumscribed about a ball in hyperbolic space, the regular simplices have minimal volume. We also investigate analogous questions for $d$-dimensional spherical and hyperbolic polytopes with $d+2$ vertices. Joint work with Zsolt Lángi.

At this session, Underrepresented Minority students and researchers will have a forum to discuss their experiences.

Some curious Brunn-Minkowski type theorems will be discussed and lead us to the study of classes of polytopes with special covering properties. Joint work with Shiri Artstein and Tomer Falah.

It is known that convex bodies in the Euclidean 3-space are rigid with respect to the induced intrinsic metric on the boundary. This story has two classical chapters: the rigidity of convex polyhedra and the rigidity of smooth convex bodies, though there is also a common generalization obtained by Pogorelov. The work of Thurston from 70s highlighted the ubiquity and the diversity of hyperbolic manifolds among 3-dimensional ones. Hyperbolic 3-manifolds with convex boundary constitute a large and interesting class to study from various perspectives. In 90’s-00’s an analogue of the Weyl problem for hyperbolic 3-manifolds with smooth convex boundary was resolved in the works of Labourie and Schlenker. Curiously enough, a polyhedral counterpart was not known until recently. One of the reasons is that some metrics on the boundary of such 3-manifolds that are «intrinsically polyhedral» admit not so polyhedral realizations, which are somewhat more difficult to handle. In my talk I will describe the state of art around these and related problems, and will present a recent proof of the respective polyhedral result in a generic case. Another outcome is a rigidity result for a family of so-called convex cocompact hyperbolic 3-manifolds, which are important in the theory of Kleinian groups. This is a step towards a resolution of conjectures of Thurston.