*Published Paper*

**Inserted:** 3 sep 2003

**Last Updated:** 24 apr 2006

**Journal:** Math. Ann.

**Volume:** 334

**Pages:** 199-233

**Year:** 2006

**Abstract:**

In the geometries of stratified groups, we show that H-convex functions locally bounded from above are locally Lipschitz continuous and that the class of v-convex functions exactly corresponds to the class of upper semicontinuous H-convex functions. As a consequence, v-convex functions are locally Lipschitz continuous in every stratified group. In the class of step 2 groups we characterize locally Lipschitz H-convex functions as measures whose distributional horizontal Hessian is positive semidefinite. In Euclidean space the same results were obtained by Dudley and Reshetnyak. We prove that a continuous H-convex function is a.e. twice differentiable whenever its second horizontal derivatives are Radon measures. Due to recent results by Garofalo-Tournier and GutiĆ©rrez-Montanari the assumptions of the previous differentiability theorem are satisfied by H-convex functions in the Heisenberg group. Then H-convex functions in the Heisenberg group satisfy the sub-Riemannian version of the classical Aleksandrov-Busemann-Feller theorem.

**Keywords:**
convex functions, lipschitz continuity, Dudley and Reshetnyak theorems, Aleksandrov theorem

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