  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.
