Towards Hadwiger's conjecture via Bourgain Slicing

Marcelo Campos; Peter van Hintum; Robert Morris; Marius Tiba

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arXiv · 2206.11227

Towards Hadwiger's conjecture via Bourgain Slicing

Marcelo Campos·Peter van Hintum·Robert Morris·Marius Tiba

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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.

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Marcelo Campos, Peter van Hintum, Robert Morris, Marius Tiba. 2022-06-22. Towards Hadwiger's conjecture via Bourgain Slicing. https://arxiv.org/abs/2206.11227

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