论文标题
朝着哈德威格的猜想通过布尔加因切片
Towards Hadwiger's conjecture via Bourgain Slicing
论文作者
论文摘要
1957年，哈德威格（Hadwiger）猜想，$ \ mathbb {r}^d $中的每个凸体都可以由$ 2^d $覆盖其内部。 60多年来，最著名的界限是$ o（4^d \ sqrt {d} \ log d）$的形式，但最近通过Huang，Slomka，Tkocz和Vritsiou提高了这一点。在本说明中，我们通过推断出Chen，Klartag和Lehec在Bourgain的切片问题上的突破性工作，迈出了迈向Hadwiger的猜想。更准确地说，我们证明，对于任何凸件$ k \ subset \ mathbb {r}^d $，$$ \ exp \ bigG（ - ω\ bigg（\ frac {d} {（\ log d} {（\ log d}^（\ log d）^8} \ bigG）我们还表明，对Bourgain的切片问题的积极答案将意味着Hadwiger的猜想有指数改善。
In 1957, Hadwiger conjectured that every convex body in $\mathbb{R}^d$ can be covered by $2^d$ translates of its interior. For over 60 years, the best known bound was of the form $O(4^d \sqrt{d} \log d)$, but this was recently improved by a factor of $e^{Ω(\sqrt{d})}$ by Huang, Slomka, Tkocz and Vritsiou. In this note we take another step towards Hadwiger's conjecture by deducing an almost-exponential improvement from the recent breakthrough work of Chen, Klartag and Lehec on Bourgain's slicing problem. More precisely, we prove that, for any convex body $K \subset \mathbb{R}^d$, $$\exp\bigg( - Ω\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.
