Width Deviation of Convex Polygons

We consider the width XT(ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_T(\omega )$$\end{document} of a convex n-gon T in the plane along the random direction ω∈R/2πZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathbb {R}}/2\pi {\mathbb {Z}}$$\end{document} and study its deviation rate: δ(XT)=E(XT2)-E(XT)2E(XT).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \delta (X_T)=\frac{\sqrt{\mathbb {E}(X^2_T)-\mathbb {E}(X_T)^2}}{\mathbb {E}(X_T)}. \end{aligned}$$\end{document}We prove that the maximum is attained if and only if T degenerates to a 2-gon. Let n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document} be an integer which is not a power of 2. We show that π4ntan(π/(2n))+π28n2sin2(π/(2n))-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sqrt{\frac{\pi }{4n\tan \hspace{0.83328pt}(\pi /(2n))}+\frac{\pi ^2}{8n^2\sin ^2(\pi /(2n))}-1} \end{aligned}$$\end{document}is the minimum of δ(XT)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta (X_T)$$\end{document} among all n-gons and determine completely the shapes of T’s which attain this minimum. They are characterized as polygonal approximations of equi-Reuleaux bodies, found and studied by Reinhardt (Jahresber. Deutsch. Math. Verein. 31, 251–270 (1922)). In particular, if n is odd, then the regular n-gon is one of the minimum shapes. When n is even, we see that regular n-gon is far from optimal. We also observe an unexpected property of the deviation rate on the truncation of the regular triangle.