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

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body ${\mathbb R}^{n}$ can be covered by a union of interiors at most its translates. Despite continuous efforts, best general upper bound for this remains it was more than sixty years ago, order ${2n \choose n}n\ln n$. In note, we improve sub-exponential factor. That is, prove n}e^{-c\sqrt{n}}$ some universal constant $c>0$. Our approach combines ideas from previous work Artstein-Avidan and second named author with tools Asymptotic Geometric Analysis. One key steps proving new lower maximum volume intersection $K$ translate $-K$; fact, get same $-K$ when they both have barycenter origin. To do so, make use measure concentration, particular thin-shell estimates isotropic log-concave measures. Using ideas, establish an exponentially better restricting our attention to bodies are $\psi_{2}$. By slightly different approach, exponential improvement established also classes positive modulus convexity.