A Characterization of the sphere and a body of revolution by means of Larman points

TL;DR

This paper investigates conditions under which convex bodies in higher dimensions are spheres or bodies of revolution, focusing on Larman points and symmetry properties, and proves several results supporting a conjecture about their characterization.

Contribution

It extends the conjecture relating Larman points to bodies of revolution from 3D to higher dimensions and proves new theorems for strictly convex origin symmetric bodies and bodies with specific symmetry conditions.

Findings

If a strictly convex origin symmetric body contains a revolution point not at the origin, it is a body of revolution.

Under certain symmetry conditions, a convex body with a Larman point is a body of revolution or a sphere.

Bodies with all hyperplane sections or projections as bodies of revolution and a unique diameter are themselves bodies of revolution.

Abstract

Let $K\subset \mathbb{R}^n$, $n\geq 3$, be a convex body. A point $p$ the interior of $K$ is said to be a 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 $p$ is a Larman point of $K$ and, in addition, for every section $\Pi\cap K$, $p$ is in the corresponding $(n-2)$-plane of symmetry, then we call $p$ a revolution point of $K$. We conjecture that if $K$ contains a Larman point which is not a revolution point, then $K$ is either an ellipsoid or a body of revolution. This generalizes a conjecture of K. Bezdek for convex bodies in $\mathbb{R}^3$ to $n \geq 4$. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if $K \subset \mathbb{R}^n$ is a strictly convex origin symmetric body that contains a revolution point $p$ which is not the origin, then $K$ is a body of revolution. This generalizes the False Axis of Revolution Theorem. We also show that if $p$ is a Larman point of $K \subset \mathbb{R}^3$ and there exists a line $L$ such that $p\notin L$ and, for every plane $\Pi$ passing through $p$, the line of symmetry of the section $\Pi \cap K$ intersects $L$, then $K$ is a body of revolution (in some cases, we conclude that $K$ is a sphere). We obtain a similar result for projections of $K$. Additionally, for $K \subset \mathbb{R}^n$, $n \geq 4$, we show that if every hyperplane section or projection of $K$ is a body of revolution and $K$ has a unique diameter $D$, then $K$ is a body of revolution with axis $D$.