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

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.