Characterizations of the sphere by means of point-projections

Abstract

In this work we prove the following: let $K$ be a convex body in the Euclidean space ${\mathbb{R}}^n$, $n\ge 3$, contained in the interior of the unit ball of ${\mathbb{R}}^n$, and let $p\in {\mathbb{R}}^n$ be a point such that, from each point of ${\mathbb{S}}^{n-1}$, $K$ looks centrally symmetric and $p$ appears as the center, then $K$ is a ball.