Characterization of the sphere by means of congruent support cones

TL;DR

This paper proves that if a convex body is enclosed by a surface where all support cones are congruent via a continuous transformation, then the body must be a sphere.

Contribution

It establishes a new characterization of spheres based on the congruence of support cones from all points on the enclosing surface.

Findings

Support cones are congruent via a continuous transformation.

The convex body enclosed by the surface is necessarily a sphere.

Provides a geometric condition for spherical symmetry.

Abstract

Let $M$ be a convex body and let $K$ be a closed convex surface $K$ both contained in the Euclidean space $\mathbb{E}^3$. What can we say about $M$ if $K$ encloses $M$ and if from all the points in $K$ the body $M$ looks the same? In this work we are going to present a result which claims that if for every two support cones $C_x$, $C_y$ of $M$, with apexes $x,y \in K$, respectively, there exists $\Phi$ in the semi direct product of the orthogonal group $O(3)$ and $\mathbb{E}^3$ such that $$C_y=\Phi(C_x),$$ and this can be done in a continuous way, then $M$ is a sphere.