by Ansgar Freyer (Freie Universität Berlin)

Europe/Rome
Aula Seminari

Aula Seminari

Description

The slicing problem due to Bourgain was, until very recently, one of the major open problems in convexity. In its most elementary form it asks whether there exists an absolute constant c>0 such that any convex body of volume 1 admits a hyperplane section through its center of volume at least c. This problem implies (or is equivalent to) a multitude of other natural conjectures in convex geometry, asymptotic analysis and geometric probability. After a series of breakthroughs it was solved affirmatively by Klartag and Lehec in December 2024.

In 2013, Koldobsky asked for a discrete version of the slicing conjecture in which the number of integer points of a convex body is to be bounded from above by the number of points in a hyperplane section. Although similar at first glance, this problem seems to be of a different, more combinatorial, nature. In particular, one cannot hope for an absolute constant.

A major challenge in the discrete setting is to deal with the lack of homogeneity of the lattice point enumerator. In the talk we present a strategy that (to a certain extend) overcomes this issue and yields a polynomial bound in Koldobsky's discrete slicing problem. Moreover, we give lower bounds on the constant in the discrete slicing problem and, if time permits, discuss a probabilistic approach due to Regev. The talk is based on joint works with Martin Henk and Eduardo Lucas.