Title: An improved bound for Hadwiger’s covering problem via thin shell inequalities for the convolution square.

Abstract: A long-standing open problem, known as Hadwiger’s covering problem, asks what is the smallest natural number N(n) such that every convex body in {\mathbb R}^n can be covered by a union of the interiors of at most N(n) of its translates. Despite continuous efforts, the best-known general upper bound for this number remain as it was more than half a decade ago, and is of the order of \binom{2n}{n}n\ln n.

In this talk, I will discuss this problem and present a new result joint with Han Huang, in which we improve this bound by a sub-exponential factor. Our approach combines ideas from previous work (in which we recovered the best to then bound) with modern tools from Asymptotic Geometric Analysis. More precisely, we prove a lower bound for the maximum of the convolution square of a convex body by comparing two log-concave probability measures that are supported on the same convex body but are concentrated around two different thin-shells.

If time permits we shall discuss some other methods and results concerning this problem and its relatives.

Abstract: A long-standing open problem, known as Hadwiger’s covering problem, asks what is the smallest natural number N(n) such that every convex body in {\mathbb R}^n can be covered by a union of the interiors of at most N(n) of its translates. Despite continuous efforts, the best-known general upper bound for this number remain as it was more than half a decade ago, and is of the order of \binom{2n}{n}n\ln n.

In this talk, I will discuss this problem and present a new result joint with Han Huang, in which we improve this bound by a sub-exponential factor. Our approach combines ideas from previous work (in which we recovered the best to then bound) with modern tools from Asymptotic Geometric Analysis. More precisely, we prove a lower bound for the maximum of the convolution square of a convex body by comparing two log-concave probability measures that are supported on the same convex body but are concentrated around two different thin-shells.

If time permits we shall discuss some other methods and results concerning this problem and its relatives.

## Date:

Wed, 24/10/2018 - 12:00 to 13:00

## Location:

Room 70, Ross building