Convex bodies with centrally symmetric sections

TL;DR

This paper proves that certain centrally symmetric convex bodies satisfying a symmetry condition with respect to an interior ball are ellipsoids, advancing the Barker-Larman conjecture in a special symmetric case.

Contribution

It establishes that centrally symmetric, strictly convex bodies satisfying the Barker-Larman condition with respect to a suitable interior ball are ellipsoids, confirming a special case of the conjecture.

Findings

Centrally symmetric bodies satisfying the condition are ellipsoids.

The result applies to strictly convex bodies with a specific interior ball.

Supports the Barker-Larman conjecture in symmetric cases.

Abstract

Let $K\subset \mathbb{R}^n$ be a convex body, $n\geq 3$. We say that $K$ satisfies the Barker-Larman condition if there exists a ball $B$ in the interior of $K$ such that for every suppor hyperplane $\Pi$ of $B$, the section $\Pi \cap K$ is a centrally symmetric set. Barker and Larman conjectured that the Barker-Larman condition characterizes the ellipsoid. In this work we prove an special case of such conjecture, in particular, we assume that the convex body $K$ is centrally symmetric. Our main result is the following: Let $K$ be a centrally symmetric and strictly convex body, with center at $O$, and let $B$ be a ball in the interior of $K$ and not containing $O$: If $K$ satisfies the Barker-Larman condition with respect to $B$ and $B$ is suitable for $K$ (intuitively, $B$ is suitable for $K$ if the boundary of $B$ is not very close to the boundary of $K$), then $K$ is an ellipsoid.