Why Study Spherical Convexity of Non-Homogeneous Quadratics and what Makes it Surprising?

Abstract This paper establishes necessary, sufficient, and equivalent conditions for the spherical convexity of non-homogeneous quadratic functions. By examining criteria for determining spherical convexity, we identified unique properties that differentiate 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, our analysis focuses on proper spherically convex subsets of the sphere. Our primary results concern non-homogeneous quadratic functions on the spherically convex set of unit vectors with positive coordinates. We also extend our findings to more general spherically convex sets. Additionally, the paper explores special cases of non-homogeneous quadratic functions where the defining matrix is of a specific type, such as positive, diagonal, or a Z-matrix. This study not only provides useful criteria for spherical convexity but also reveals surprising characteristics of spherically convex quadratic functions, contributing to a deeper understanding of convexity in spherical geometries.