Communications in Analysis and Geometry

Volume 12 (2004)

Number 4

The Theorem of Busemann-Feller-Alexandrov in Carnot Groups

Pages: 853 – 886

DOI: https://dx.doi.org/10.4310/CAG.2004.v12.n4.a5

Authors

D. Danielli

N. Garofalo

D. M. Nhieu

F. Tournier

Abstract

A classical theorem states that a convex function in ℝⁿ admits second derivatives at a.e. point. This result was first proved by Busemann and Feller [BF] for functions in the plane, and subsequently generalized by A.D. Alexandrov [A] to arbitrary dimensions. The theorem of Busemann-Feller-Alexandrov plays a basic role in analysis and in pde's, especially in the theory of fully nonlinear equations. For instance, in the proof of uniqueness of viscosity solutions, see Theorems 5.1 and 5.3 in [CC], a quantitative version of such result (see Theorem 6.4.1 in [EG]) plays an essential role. In this paper we prove a version of the Busemann-Feller-Alexandrov theorem for the class of weakly H-convex functions in Carnot groups introduced in [DGN].

Published 1 January 2004