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Journal of Convex Analysis 21 (2014), No. 3, 785--810
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

Federica Dragoni
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 2AG Wales, England

We study properties of functions convex with respect to a given family χ of vector fields, a notion that appears natural in Carnot-Carathéodory metric spaces. We define a suitable subdifferential and show that a continuous function is χ-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 χ-affine functions and show their connections with χ-convexity.

Keywords: Convex functions in Carnot groups, Carnot-Caratheodory metric spaces, subdifferential, Legendre-Fenchel transform, convex duality, Jensen inequality.

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