Subdifferential and Properties of Convex Functions with Respect to Vector Fields

We study properties of functions convex with respect to a given family $\mathcal{X}$ 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 $\mathcal{X}$-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 $\mathcal{X}$-affine functions and show their connections with $\mathcal{X}$-convexity.