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.
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