Convexity of sets and quadratic functions on the hyperbolic space

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. Then, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex function and present the first and second order characterizations for these functions. An extensive study of the hyperbolic convex quadratic function is also presented.