Convex bodies of constant width with exponential illumination number

Andrii Arman; Andriy Bondarenko; Andriy Prymak

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arXiv · 2304.10418

Convex bodies of constant width with exponential illumination number

Andrii Arman·Andriy Bondarenko·Andriy Prymak

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Abstract

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 G. 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 by J. Bourgain and J. Lindenstrauss.

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Andrii Arman, Andriy Bondarenko, Andriy Prymak. 2023-04-20. Convex bodies of constant width with exponential illumination number. https://arxiv.org/abs/2304.10418

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