# Improved bounds for Hadwiger’s covering problem via thin-shell estimates

### Han Huang

University of Michigan, Ann Arbor, USA### Boaz A. Slomka

Weizmann Institute of Science, Rehovot, Israel### Tomasz Tkocz

Carnegie Mellon University, Pittsburgh, USA### Beatrice-Helen Vritsiou

University of Alberta, Edmonton, Canada

## Abstract

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N(n)$ is 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 general upper bound known for this number remains as it was more than sixty years ago, of the order of ${2n \choose n}n\ln n$.

In this note, we improve this bound by a subexponential factor. That is, we prove a bound of the order of ${2n \choose n}e^{-c\sqrt{n}}$ for some universal constant $c>0$.

Our approach combines ideas from [3] by Artstein-Avidan and the second named author with tools from asymptotic geometric analysis. One of the key steps is proving a new lower bound for the maximum volume of the intersection of a convex body $K$ with a translate of $-K$; in fact, we get the same lower bound for the volume of the intersection of $K$ and $-K$ when they both have barycenter at the origin. To do so, we make use of measure concentration, and in particular of thin-shell estimates for isotropic log-concave measures.

Using the same ideas, we establish an exponentially better bound for $N(n)$ when restricting our attention to convex bodies that are $\psi_{2}$. By a slightly different approach, an exponential improvement is established also for classes of convex bodies with positive modulus of convexity.

## Cite this article

Han Huang, Boaz A. Slomka, Tomasz Tkocz, Beatrice-Helen Vritsiou, Improved bounds for Hadwiger’s covering problem via thin-shell estimates. J. Eur. Math. Soc. 24 (2022), no. 4, pp. 1431–1448

DOI 10.4171/JEMS/1132