具有指数照明数的恒宽凸面体

Pub Date : 2024-05-04 DOI:10.1007/s00454-024-00647-9

Andrii Arman, Andrii Bondarenko, Andriy Prymak

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引用次数: 0

摘要

我们证明了在\({\mathbb {E}}^n\) 中存在照明数至少为 \((\cos (\pi /14)+o(1))^{-n}\)的恒宽凸体，这回答了卡莱提出的一个问题。此外，我们证明了在\({\mathbb {E}}^n\) 中存在直径为 1 的有限集合，这些集合不能被直径为 1 的球（（2/\sqrt{3}-o(1)）^{n}\）覆盖，从而改进了布尔甘和林登斯特劳斯的一个结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

Convex Bodies of Constant Width with Exponential Illumination Number

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Convex Bodies of Constant Width with Exponential Illumination Number

We show that there exist convex bodies of constant width in \({\mathbb {E}}^n\) with illumination number at least \((\cos (\pi /14)+o(1))^{-n}\), answering a question by Kalai. Furthermore, we prove the existence of finite sets of diameter 1 in \({\mathbb {E}}^n\) which cannot be covered by \((2/\sqrt{3}-o(1))^{n}\) balls of diameter 1, improving a result of Bourgain and Lindenstrauss.

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