Characterizations of the sphere by means of visual cones: an alternative proof of Matsuura's theorem

Abstract

In this work we prove that if there exists a smooth convex body $M$ in the Euclidean space $\mathbb{R}^n$, $n\geq 3$, contained in the interior of the unit ball $\mathbb{S}^{n-1}$ of $\mathbb{R}^n$, and point $p\in \mathbb{R}^n$ such that, for each point of $\mathbb{S}^{n-1}$, $M$ looks centrally symmetric and $p$ appears as the centre, then $M$ is an sphere