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

Abstract

In this work we prove the following result: Let $K$ be a strictly convex body in the Euclidean space ${\mathbb{R}}^n,n\ge 3$, 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.

References in corpus (1)

• Convex Bodies with affinely equivalent projections and affine bodies of revolution

Cited by in corpus (2)

• On characteristic properties of the ellipsoid in terms of circumscribed cones of a convex body
• Characterization of the sphere by means of congruent support cones