DOI: 10.68381/jca21042 ISSN: 0944-6532
Subdifferential and Properties of Convex Functions with Respect to Vector Fields
Martino Bardi, Federica Dragoni
We study properties of functions convex with respect to a given family
\mathcal{X}
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}
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}
X
-affine functions and show their connections with
\mathcal{X}
X
-convexity.