Maximal perimeters of polytope sections and origin-symmetry

Abstract

Let $P\subset {\mathbb{R}}^n$ $(n\ge 3)$ be a convex polytope containing the origin in its interior. Let $\mbox{vol}_{n-2} \big( \mbox{relbd} ( P\cap\lbrace tξ+ ξ^\perp \rbrace ) \big)$ denote the $(n-2)$-dimensional volume of the relative boundary of $P\cap \{t\xi +{\xi }^{\perp }\}$ for $t\in \mathbb{R}$, $\xi \in S^{n-1}$. We prove the following: if \begin{align*} \mbox{vol}_{n-2} \Big( \mbox{relbd} \big( P\capξ^\perp \big) \Big) = \max_{t\in\mathbb{R}} \mbox{vol}_{n-2} \Big( \mbox{relbd} \big( P\cap\lbrace tξ+ ξ^\perp \rbrace \big) \Big) \ \ \forall \ \ ξ\in S^{n-1}, \end{align*} then $P$ is origin-symmetric, i.e. $P=-P$. Our result gives a partial affirmative answer to a conjecture by Makai, Martini, and Ódor. We also characterize the origin-symmetry of $C^1$ convex bodies in terms of the dual quermassintegrals of their sections; this can be seen as a dual version of the conjecture of Makai et al.

14 pages, 2 figures