Convexity of sets and quadratic functions on the hyperbolic space

TL;DR

This paper explores convex analysis in hyperbolic space, including properties of the intrinsic distance, convex sets, functions, and quadratic functions, with theoretical characterizations and optimization insights.

Contribution

It provides new theoretical insights into convexity, intrinsic distances, and quadratic functions specifically within hyperbolic geometry, extending convex analysis concepts.

Findings

Spectral decomposition of the intrinsic distance Hessian

Characterizations of convex functions in hyperbolic space

Analysis of hyperbolically convex quadratic functions

Abstract

In this paper some concepts of convex analysis on hyperbolic space are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex functions and present first and second order characterizations of these functions, as well as some optimization concepts related to them. An extensive study of the hyperbolically convex quadratic functions is also presented.