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Subdifferential and Properties of Convex Functions with Respect to Vector Fields

Abstract : 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.
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https://hal.inria.fr/hal-00916081
Contributor : Estelle Bouzat <>
Submitted on : Monday, December 9, 2013 - 5:05:45 PM
Last modification on : Thursday, July 4, 2019 - 3:50:02 PM

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  • HAL Id : hal-00916081, version 1

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Martino Bardi, Federica Dragoni. Subdifferential and Properties of Convex Functions with Respect to Vector Fields. Journal of Convex Analysis, Heldermann, 2014, 21 (3), pp.785--810. ⟨hal-00916081⟩

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