Convex Functions on Sub-Riemannian Manifolds. I

Abstract

We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofalo-Nieuhn (equivalent to that by Lu-Manfredi-Stroffolini). Nonholonomic geodesics are defined using the horizontal connection. A new distance corresponding to the horizontal connection has been introduced and near regular points proven to be equivalent to the Carnot-Carath\`{e}odory distance. Some basic properties of convex functions are studied. In particular we prove that any nonholonomically geodesic convex function locally bounded from above is locally Lipschitzian with respect to the Carnot-Carath\`{e}odory distance.