Binary Operations in Spherical Convex Geometry

TL;DR

This paper characterizes binary operations on convex bodies on the sphere, showing the convex hull is the unique non-trivial projection covariant operation, and that continuous, projection covariant operations are trivial.

Contribution

It establishes the uniqueness of the convex hull as a non-trivial projection covariant operation on spherical convex bodies and proves the triviality of continuous, projection covariant operations.

Findings

Convex hull is the only non-trivial projection covariant operation.

Continuous and projection covariant operations are trivial.

Characterizations are specific to convex bodies in open hemispheres.

Abstract

Characterizations of binary operations between convex bodies on the Euclidean unit sphere are established. The main result shows that the convex hull is essentially the only non-trivial projection covariant operation between pairs of convex bodies contained in open hemispheres. Moreover, it is proved that any continuous and projection covariant binary operation between all proper spherical convex bodies must be trivial.