Convex Bodies of Constant Width with Exponential Illumination Number

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