Convexity and semiconvexity along vector fields

Abstract

Given a family of vector fields we introduce a notion of convexity and of semiconvexity of a function along the trajectories of the fields and give infinitesimal characterizations in terms of inequalities in viscosity sense for the matrix of second derivatives with respect to the fields. We also prove that such functions are Lipschitz continuous with respect to the Carnot–Caratheodory distance associated to the family of fields and have a bounded gradient in the directions of the fields. This extends to Carnot–Caratheodory metric spaces several results for the Heisenberg group and Carnot groups obtained by a number of authors.