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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 aquestion by G. Kalai. Furthermore, we prove the existence of finite sets ofdiameter $1$ 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.

Convex bodies of constant width with exponential illumination number