
Journal of Convex Analysis 21 (2014), No. 3, 785810 Copyright Heldermann Verlag 2014 Subdifferential and Properties of Convex Functions with Respect to Vector Fields Martino Bardi Dip. di Matematica, Università di Padova, via Trieste 63, 35121 Padova, Italy bardi@math.unipd.it Federica Dragoni School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 2AG Wales, England DragoniF@cardiff.ac.uk We study properties of functions convex with respect to a given family χ of vector fields, a notion that appears natural in CarnotCarathéodory metric spaces. We define a suitable subdifferential and show that a continuous function is χconvex 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 χaffine functions and show their connections with χconvexity. Keywords: Convex functions in Carnot groups, CarnotCaratheodory metric spaces, subdifferential, LegendreFenchel transform, convex duality, Jensen inequality. [ Fulltextpdf (217 KB)] for subscribers only. 