# Recent Advances in Discrete and Analytic Aspects of Convexity (17w5074)

Arriving in Banff, Alberta Sunday, May 21 and departing Friday May 26, 2017

## Organizers

Karoly Bezdek (University of Calgary)

Alexander Koldobsky (University of Missouri)

Dmitry Ryabogin (Kent State University)

Artem Zvavitch (Kent State University)

Vladyslav Yaskin (University of Alberta)

## Objectives

The main goal of the proposed workshop at BIRS would be to bring together the leading experts and young researchers in the area to coordinate the new applications of Harmonic Analysis, Discrete Geometry and Symplectic Geometry to Convexity.

The proposed workshop will focus on the mixture of several topics in discrete and analytic Convexity, but will also have very close relation to Harmonic, Geometric and Functional Analysis.

Below we outline two main directions of the workshop that represent research interests of the majority of the proposed participants. We include the list of problems related to these directions.

1. Uniqueness results and inequalities for sections and projections and their discrete analogues.

1-A. The discrete analogue of Alexandrov problem about unique determination of origin-symmetric convex bodies by the areas of their orthogonal projections (here we replace the bodies with discrete sets, obtained as the the intersection of bodies with the grid of the corresponding dimension, and the areas of projections with the number of points projected onto the corresponding plane).

1-A. A classical theorem of Alexandrov states that origin-symmetric convex bodies are uniquely determined by the areas of their projections. Is there a version of this result in discrete settings? (Replace bodies by lattice sets, and the areas of projections by the counting measure).

1-B. An old open problem of Zuss asks whether two convex bodies having pairwise congruent orthogonal projections onto the corresponding planes must coincide up to translation and reflection in the origin.

1-C. A problem of Bezdek is whether a convex body with all its sections having axes of symmetry must be an ellipsoid or a body of revolution.

1-D. Modifications and generalizations of the Busemann-Petty problem. One of them is the well-known slicing problem, which asks whether any convex body of volume one has a large enough section (larger than a fixed constant independent of anything). Another is a question of Koldobsky about a discrete analogue of the slicing problem.

1-E. Questions about duality and the Mahler conjecture. A number of recent results of Artstein-Avidan, Karasev and Ostrover (2013) on billiards as well as on generated algorithms to compute the Ekeland-Hofer-Zehnder capacities has found new connection between the longstanding Mahler conjecture in convexity and the conjecture of Viterbo in symplectic geometry.

Another interesting problem that might be related to the theory of billiards is the so-called t-sections problem. It goes back to Santalo and asks the following. Suppose that two convex bodies containing the ball of radius t in their interior are such that their corresponding sections by the hyperplanes tangent to the ball are of the same volume. Does it follow that $K=L$? This problem is open even for plane bodies.

2. Extremal problems in convex and discrete geometry and questions that are coming to discrete geometry from analysis.

2-A: A century old conjecture - Are the Meissner's bodies minimizing the volume of convex bodies of given constant width in Euclidean 3-space?

2-B: Boltyanski-Hadwiger illumination conjecture: prove that any d-dimensional convex body can be illuminated by $2^d$ light sources in Euclidean d-space and $2^d$ is needed only for the affine d-cube.

2-C: Contact number problem (generalizing the problem of kissing numbers): find the largest number of touching pairs (contacts) in a packing of n unit balls in Euclidean d-space.

2-D: Zelditch problem on characterizing convex bodies whose shortest periodic billiard trajectories are of period 2.

2-E: Foam problems such as the following: If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells?

2-F: Kneser-Poulsen conjecture (1955): prove that the volume of the union of finitely many balls decreases under any (discrete) contraction of the center points in Eucldean d-space.

2-G: Alexander conjecture (1985): Under arbitrary contraction of the center points of finitely many congruent disks in the Euclidean plane, the perimeter of the intersection of the disks cannot decrease.

All these topics are being actively pursued at the present time and any progress in these directions would be a tremendous success of the workshop. We believe it is the right time for such a meeting to take place.

The proposed workshop will focus on the mixture of several topics in discrete and analytic Convexity, but will also have very close relation to Harmonic, Geometric and Functional Analysis.

Below we outline two main directions of the workshop that represent research interests of the majority of the proposed participants. We include the list of problems related to these directions.

1. Uniqueness results and inequalities for sections and projections and their discrete analogues.

1-A. The discrete analogue of Alexandrov problem about unique determination of origin-symmetric convex bodies by the areas of their orthogonal projections (here we replace the bodies with discrete sets, obtained as the the intersection of bodies with the grid of the corresponding dimension, and the areas of projections with the number of points projected onto the corresponding plane).

1-A. A classical theorem of Alexandrov states that origin-symmetric convex bodies are uniquely determined by the areas of their projections. Is there a version of this result in discrete settings? (Replace bodies by lattice sets, and the areas of projections by the counting measure).

1-B. An old open problem of Zuss asks whether two convex bodies having pairwise congruent orthogonal projections onto the corresponding planes must coincide up to translation and reflection in the origin.

1-C. A problem of Bezdek is whether a convex body with all its sections having axes of symmetry must be an ellipsoid or a body of revolution.

1-D. Modifications and generalizations of the Busemann-Petty problem. One of them is the well-known slicing problem, which asks whether any convex body of volume one has a large enough section (larger than a fixed constant independent of anything). Another is a question of Koldobsky about a discrete analogue of the slicing problem.

1-E. Questions about duality and the Mahler conjecture. A number of recent results of Artstein-Avidan, Karasev and Ostrover (2013) on billiards as well as on generated algorithms to compute the Ekeland-Hofer-Zehnder capacities has found new connection between the longstanding Mahler conjecture in convexity and the conjecture of Viterbo in symplectic geometry.

Another interesting problem that might be related to the theory of billiards is the so-called t-sections problem. It goes back to Santalo and asks the following. Suppose that two convex bodies containing the ball of radius t in their interior are such that their corresponding sections by the hyperplanes tangent to the ball are of the same volume. Does it follow that $K=L$? This problem is open even for plane bodies.

2. Extremal problems in convex and discrete geometry and questions that are coming to discrete geometry from analysis.

2-A: A century old conjecture - Are the Meissner's bodies minimizing the volume of convex bodies of given constant width in Euclidean 3-space?

2-B: Boltyanski-Hadwiger illumination conjecture: prove that any d-dimensional convex body can be illuminated by $2^d$ light sources in Euclidean d-space and $2^d$ is needed only for the affine d-cube.

2-C: Contact number problem (generalizing the problem of kissing numbers): find the largest number of touching pairs (contacts) in a packing of n unit balls in Euclidean d-space.

2-D: Zelditch problem on characterizing convex bodies whose shortest periodic billiard trajectories are of period 2.

2-E: Foam problems such as the following: If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells?

2-F: Kneser-Poulsen conjecture (1955): prove that the volume of the union of finitely many balls decreases under any (discrete) contraction of the center points in Eucldean d-space.

2-G: Alexander conjecture (1985): Under arbitrary contraction of the center points of finitely many congruent disks in the Euclidean plane, the perimeter of the intersection of the disks cannot decrease.

All these topics are being actively pursued at the present time and any progress in these directions would be a tremendous success of the workshop. We believe it is the right time for such a meeting to take place.