Maximal perimeters of polytope sections and origin-symmetry

Abstract

Let $P\subset\mathbb{R}^n$ $(n\geq 3)$ be a convex polytope containing the origin in its interior. Let $\mbox{vol}_{n-2} \big( \mbox{relbd} ( P\cap\lbrace t\xi + \xi^\perp \rbrace ) \big)$ denote the $(n-2)$-dimensional volume of the relative boundary of $P\cap\lbrace t\xi + \xi^\perp \rbrace$ 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\cap\xi^\perp \big) \Big) = \max_{t\in\mathbb{R}} \mbox{vol}_{n-2} \Big( \mbox{relbd} \big( P\cap\lbrace t\xi + \xi^\perp \rbrace \big) \Big) \ \ \forall \ \ \xi\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 \'Odor. 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.