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: $$ \delta(X_T)=\frac{\sqrt{\mathbb{E}(X^2_T)-\mathbb{E}(X_T)^2}}{\mathbb{E}(X_T)}. $$ We prove that the maximum is attained if and only if $T$ degenerates to a $2$-gon. Let $n\geq 2$ be an integer which is not a power of $2$. We show that $$ \sqrt{\frac{\pi}{4n\tan(\frac{\pi}{2n})} +\frac{\pi^2}{8n^2\sin^2(\frac{\pi}{2n})}-1} $$ 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 K.~Reinhardt. 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.