Calculus of Variations and Geometric Measure Theory

M. Bardi - F. Dragoni

Subdifferential and Properties of Convex Functions with respect to Vector Fields

created by bardi on 17 Jul 2013
modified on 07 Jan 2015


Published Paper

Inserted: 17 jul 2013
Last Updated: 7 jan 2015

Journal: J. Convex Analysis
Volume: 21
Number: 3
Pages: 785--810
Year: 2014


We study properties of functions convex with respect to a given family X of vector fields, a notion that appears natural in Carnot-Caratheodory metric spaces. We define a suitable subdifferential and show that a continuous function is X-convex if and only if such subdifferential is nonempty at every point. For vector fields of Carnot type we deduce from this property that a generalized Fenchel transform is involutive and a weak form of Jensen inequality. Finally we introduce and compare several notions of X-affine functions and show their connections with X-convexity.