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

TL;DR

The paper addresses a visual-cone based characterization of central symmetry for convex bodies in $\mathbb{R}^n$ with $n\geq 3$. It studies strictly convex $K$ and a hypersurface $L$ containing $K$ in its interior, imposing that for every apex $x\in L$ there exists $y\in L$ and a translation such that the corresponding support double-cones $C_x$ and $C_y$ coincide up to translation. The authors develop a sequence of geometric lemmas about the intersections of cones with $\operatorname{bd}K$, inverse homothety relations between cone sections, and a continuity argument to extend local symmetry to all of $L$, ultimately proving that $K$ and $L$ are centrally symmetric and concentric. This provides a rigorous, higher-dimensional, cone-driven criterion for central symmetry with potential implications for isoptic-type and projection-based convexity problems. All mathematical notation is used with $...$ to clearly delimit the involved expressions.

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.

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Theorems & Definitions (9)

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