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 Rn 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 (2nn)n ln n. In this note, we improve this bound by a subexponential factor. That is, we prove a bound of the order of (2nn)e-c√n for some universal constant c > 0. Our approach combines ideas from  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 Ψ2. By a slightly different approach, an exponential improvement is established also for classes of convex bodies with positive modulus of convexity.
|Number of pages||18|
|Journal||Journal of the European Mathematical Society|
|State||Published - 2021|
Bibliographical noteFunding Information:
Funding. We thank the Mathematical Sciences Research Institute in Berkeley, California, where part of this work was done during the Fall 2017 semester; we acknowledge the support of the institute and of the National Science Foundation under Grant No. 1440140. TT was supported in part by the Collaboration Grants from the Simons Foundation and NSF grant DMS-1955175.
© 2021 European Mathematical Society.
- Convex body
- Covering number
- Measure of symmetry