Why Study the Spherical Convexity of Non-Homogeneous Quadratic Functions, and What Makes It Surprising?

TL;DR

This paper characterizes the spherical convexity of non-homogeneous quadratic functions, revealing surprising distinctions from geodesic convexity and providing criteria for various quadratic forms on convex subsets of the sphere.

Contribution

It establishes necessary and sufficient conditions for spherical convexity of non-homogeneous quadratic functions and uncovers unexpected properties differentiating them from geodesically convex functions.

Findings

Conditions for spherical convexity of quadratic functions

Surprising differences from geodesic convexity in hyperbolic and Euclidean spaces

Results for specific matrix types like positive, diagonal, and Z-matrices

Abstract

This paper presents necessary, sufficient, and equivalent conditions for the spherical convexity of non-homogeneous quadratic functions. In addition to motivating this study and identifying useful criteria for determining whether such functions are spherically convex, we discovered surprising properties that distinguish spherically convex quadratic functions from their geodesically convex counterparts in both hyperbolic and Euclidean spaces. Since spherically convex functions over the entire sphere are constant, we restricted our focus to proper spherically convex subsets of the sphere. Although most of our results pertain to non-homogeneous quadratic functions on the spherically convex set of unit vectors with positive coordinates, we also present findings for more general spherically convex sets. Beyond the general non-homogeneous quadratic functions, we consider explicit special cases where the matrix in the function's definition is of a specific type, such as positive, diagonal, and Z-matrix.