Convex Bodies of Constant Width with Exponential Illumination Number

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.