Convex bodies with sections with hyperplanes of symmetry

Abstract

Let $K\subset \mathbb{R}^n$ be a convex body and let $p$ in the interior of $ K$, $n \geq 3$. The point $p$ is said to be a \textit{Larman point} of $K$ if, for every hyperplane $\Pi$ passing through $p$, the section $\Pi\cap K$ has a $(n-2)$-plane of symmetry. If, in addition, for every hyperplane $\Pi$ passing through $p$, the section $\Pi\cap K$ has a $(n-2)$-plane of symmetry which contains $p$, then the point $p$ is called a revolution point. In this work we prove that if for the convex body $K$, $n \geq 3$, there exists a hyperplane $H$, a point $p$ such that $p$ is a Larman point of $K$ but not a revolution point and, for every hyperplane $\Pi$ passing though $p$, the section $\Pi \cap K$ has an $(n-2)$-plane of symmetry parallel to $H$, then $K$ is an ellipsoid of revolution with an axis perpendicular to $H$.