A characterization of centrally symmetric convex bodies in terms of visual cones

TL;DR

This paper characterizes when a strictly convex body in Euclidean space is centrally symmetric based on the properties of support double-cones with respect to a hypersurface containing it.

Contribution

It provides a new geometric characterization of central symmetry for convex bodies using support double-cones and hypersurfaces.

Findings

Convex body $K$ is centrally symmetric if support double-cones differ by translation.

Hypersurface $L$ containing $K$ must be centrally symmetric and concentric with $K$.

The geometric condition on support double-cones characterizes symmetry.

Abstract

In this work we prove the following result: Let $K$ be a strictly convex body in the Euclidean space $\mathbb{R}^n, n\geq 3$, and let $L$ be a hypersurface, which is the image of an embedding of the sphere $\mathbb{S}^{n-1}$, such that $K$ is contained in the interior of $L$. Suppose that, for every $x\in L$, there exists $y\in L$ such that the support double-cones of $K$ with apexes at $x$ and $y$, differ by a translation. Then $K$ and $L$ are centrally symmetric and concentric.