2007/01/10 by Kang-Hai Tan, Tan, Kang-Hai
Mathematics · #53C17 #58C05 #58C99 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.math/0701273
openalex publication_date 2007/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-Carathèodory 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-Carathèodory distance.