Towards Hadwiger's conjecture via Bourgain Slicing

TL;DR

This paper advances Hadwiger's conjecture by leveraging recent progress on Bourgain's slicing problem, showing that a near-exponential number of translates of a convex body's interior can cover the body, with implications for future breakthroughs.

Contribution

It establishes an almost-exponential bound on the number of translates needed to cover convex bodies, connecting Bourgain's slicing problem to Hadwiger's conjecture.

Findings

Proves a near-exponential bound for covering convex bodies with translates of their interior.

Shows that solving Bourgain's slicing problem positively would exponentially improve Hadwiger's conjecture.

Builds a bridge between Bourgain's slicing problem and geometric covering problems.

Abstract

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 $e^{\Omega(\sqrt{d})}$ 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 $K \subset \mathbb{R}^d$, $$\exp\bigg( - \Omega\bigg( \frac{d}{(\log d)^8} \bigg) \bigg) \cdot 4^d$$ 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.