Convex Bodies of Constant Width with Exponential Illumination Number

We show that there exist convex bodies of constant width in En\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {E}}<^>n$$\end{document} with illumination number at least (cos(pi/14)+o(1))-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\cos (\pi /14)+o(1))<^>{-n}$$\end{document}, answering a question by Kalai. Furthermore, we prove the existence of finite sets of diameter 1 in En\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {E}}<^>n$$\end{document} which cannot be covered by (2/3-o(1))n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2/\sqrt{3}-o(1))<^>{n}$$\end{document} balls of diameter 1, improving a result of Bourgain and Lindenstrauss.