Geodesic
Subdifferential and Properties of Convex Functions with Respect to Vector Fields
Journal of convex analysis, Tome 21 (2014) no. 3, pp. 785-81.

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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.
Mots-clés : Convex functions in Carnot groups, Carnot-Caratheodory metric spaces, subdifferential, Legendre-Fenchel transform, convex duality, Jensen inequality 
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     author = {M. Bardi and F. Dragoni},
     title = {Subdifferential and {Properties} of {Convex} {Functions} with {Respect} to {Vector} {Fields}},
     journal = {Journal of convex analysis},
     pages = {785--81},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2014},
     url = {http://geodesic.mathdoc.fr/item/JCA_2014_21_3_JCA_2014_21_3_a10/}
}
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M. Bardi; F. Dragoni. Subdifferential and Properties of Convex Functions with Respect to Vector Fields. Journal of convex analysis, Tome 21 (2014) no. 3, pp. 785-81. http://geodesic.mathdoc.fr/item/JCA_2014_21_3_JCA_2014_21_3_a10/