Characterizations of the sphere by means of point-projections

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