Width Deviation of Convex Polygons

Abstract

We consider the width $X_T(\omega )$ of a convex n-gon T in the plane along the random direction $\omega \in {\mathbb {R}}/2\pi {\mathbb {Z}}$ and study its deviation rate:

$$\begin{aligned} \delta (X_T)=\frac{\sqrt{\mathbb {E}(X^2_T)-\mathbb {E}(X_T)^2}}{\mathbb {E}(X_T)}. \end{aligned}$$

We prove that the maximum is attained if and only if T degenerates to a 2-gon. Let $n\ge 2$ be an integer which is not a power of 2. We show that

$$\begin{aligned} \sqrt{\frac{\pi }{4n\tan \hspace{0.83328pt}(\pi /(2n))}+\frac{\pi ^2}{8n^2\sin ^2(\pi /(2n))}-1} \end{aligned}$$

is the minimum of $\delta (X_T)$ 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.