Towards Hadwiger's conjecture via Bourgain Slicing

In 1957, Hadwiger conjectured that every convex body in ${\mathbb{R}}^d$ can be covered by $2^d$ translates of its interior. For over 60 years, the best known bound was of the form $O(4^d\sqrt{d}\log d)$, but this was recently improved by a factor of $e^{\mathrm{\Omega }(\sqrt{d})}$ by Huang, Slomka, Tkocz and Vritsiou. In this note we take another step towards Hadwiger’s conjecture by deducing an almost-exponential improvement from the recent breakthrough work of Chen, Klartag and Lehec on Bourgain’s slicing problem. More precisely, we prove that, for any convex body $K\subset {\mathbb{R}}^d$, $$\exp (-\mathrm{\Omega }(\frac{d}{(\log d{)}^8}\left)\right)\cdot 4^d$$ translates of $\text{int}(K)$ suffice to cover $K$. We also show that a positive answer to Bourgain’s slicing problem would imply an exponential improvement for Hadwiger’s conjecture.