Spherical Separation Theorem

TL;DR

This paper establishes a spherical separation theorem characterizing the disjointness of convex subsets on the sphere through the existence of a convex separating set.

Contribution

It introduces a new spherical separation theorem linking disjoint convex sets with a convex separating subset on the sphere.

Findings

Disjoint convex sets on the sphere can be separated by a convex set.

The separation condition is characterized by the existence of a specific convex subset.

The theorem applies to both open and closed spherical convex sets.

Abstract

In this paper, it is shown that for any two non-empty closed (resp., open) and spherical convex subsets $\mathcal{W}_1, \mathcal{W}_2$ of $S^n$, the intersection $\mathcal{W}_1\cap \mathcal{W}_2$ is empty if and only if the subset $\{P\in S^n\; |\; P\cdot Q>0 \mbox{ for any } Q\in \mathcal{W}_1 \mbox{ and } P\cdot R<0 \mbox{ for any } R\in \mathcal{W}_2\}$ is non-empty, open (resp., closed) and spherical convex.