Characterization of the sphere by means of congruent support cones

TL;DR

Problem: characterize convex bodies $M\subset \operatorname{int} K$ in $\mathbb{E}^3$ from the condition that all support cones $C_x$ with apexes $x\in \operatorname{bd} K$ are affinely congruent in a continuous way. The authors leverage fiber-bundle topology, showing the principal bundle $\tau: \mathbb{S}^2 \to O(3)$ is nontrivial, which forces the cones to be straight circular or have a symmetry that yields a field of congruent bodies. They then apply Mani's proposition to deduce a circular base section and Matsuura's theorem to conclude that $M$ is a Euclidean ball. The work thus characterizes spheres via congruent supporting cones and highlights a deep link between convex geometry and the topology of fiber bundles.

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 $Φ$ in the semi direct product of the orthogonal group $O(3)$ and $\mathbb{E}^3$ such that $$C_y=Φ(C_x),$$ and this can be done in a continuous way, then $M$ is a sphere.

Figures (1)

• Figure 1: The sequence $\{u_n\}$, $\{v_n\}$ converges to $u^*$ when $n\rightarrow \infty$.

Theorems & Definitions (3)

• Theorem 1
• Lemma 1
• proof