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: $$\delta (X_T)=\frac{\sqrt{\mathbb{E}(X_T^2)-\mathbb{E}(X_T{)}^2}}{\mathbb{E}(X_T)}.$$ We prove that the maximum is attained if and only if $T$ degenerates to a $2$-gon. Let $n\ge 2$ 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.

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