Convex bodies of constant width with exponential illumination number

We show that there exist convex bodies of constant width in ${\mathbb{E}}^n$ with illumination number at least $(\cos (\pi \mathrm{/}14)+o(1){)}^{-n}$, 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 $(2\mathrm{/}\sqrt{3}+o(1){)}^n$ balls of diameter $1$, improving a result by J. Bourgain and J. Lindenstrauss.