*Published Paper*

**Inserted:** 17 jul 2013

**Last Updated:** 7 jan 2015

**Journal:** J. Convex Analysis

**Volume:** 21

**Number:** 3

**Pages:** 785--810

**Year:** 2014

**Abstract:**

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.

**Download:**