Characterizations of the sphere by means of point-projections

TL;DR

This paper proves that if a convex body in Euclidean space appears centrally symmetric from every point on a sphere with a fixed center point, then the body must be a sphere.

Contribution

It establishes a new characterization of spheres based on point-projections and symmetry properties of convex bodies.

Findings

Convex bodies with symmetric projections from a fixed point are spheres.

The result holds in Euclidean spaces of dimension three and higher.

Provides a geometric criterion for identifying spheres.

Abstract

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.