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 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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A. Arman was supported by a postdoctoral fellowship of the Pacific Institute of Mathematical Sciences and the Department of Mathematics of the University of Manitoba. A. Bondarenko was supported in part by Grant 334466 of the Research Council of Norway. A. Prymak was supported by NSERC of Canada Discovery Grant RGPIN-2020-05357.
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Arman, A., Bondarenko, A. & Prymak, A. Convex Bodies of Constant Width with Exponential Illumination Number. Discrete Comput Geom (2024). https://doi.org/10.1007/s00454-024-00647-9
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DOI: https://doi.org/10.1007/s00454-024-00647-9