• carnot group
• h-convex
• stratified groups

The present thesis deals with a number of geometric properties of convex functions in a non-Euclidean framework. This setting is represented by the so-called Sub-Riemannian space, also called a Carnot-Caratheodory (CC) space, that can be thought of as a space where the metric structure is a constrained geometry and one can move only along a prescribed set of directions depending on the point. We will study first order and second order reguarity of h-convex functions using h-subdifferentials. Moreover the distributional notion of h-convexity is considered. In this context we will prove that for all stratified groups an h-convex distribution is represented by an h-convex function. In the last chapter we address the study of convexity in general CC space. Here we prove a quantitative Lipschitz estimate for convex functions.