Abstract

Given a real-valued function cc defined on the cartesian product of a generic Carnot group G\G and the first layer V1V_1 of its Lie algebra, we introduce a notion of cc horizontal convex (cc H-convex) function on G\G as the supremum of a suitable family of affine functions; this family is defined pointwisely, and depends strictly on the horizontal structure of the group. This abstract approach provides cc H-convex functions that, under appropriate assumptions on c,c, are characterized by the nonemptiness of the cc H-subdifferential and, above all, are locally H-semiconvex, thereby admitting horizontal derivatives almost everywhere. It is noteworthy that such functions can be recovered via a Rockafellar technique, starting from a suitable notion of cc H-cyclic monotonicity for maps. In the particular case where c(g,v)=⟨ξ1(g),v⟩,c(g,v)=\langle \xi_1(g),v \rangle, we obtain the well-known weakly H-convex functions introduced by Danielli, Garofalo and Nhieu. Finally, we suggest a possible application to optimal mass transportation.

Author information

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Andrea Calogero

Dipartimento di Statistica, Università degli Studi di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy

andrea.calogero@unimib.it

Rita Pini

Dipartimento di Statistica, Università degli Studi di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy

rita.pini@unimib.it

Keywords

  • Carnot group
  • horizontal convexity
  • c horizontal convexity
  • c horizontal differential
  • c horizontal cyclic monotonicity

Mathematics Subject Classification

52A01; 22E25

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Suggested citation

A. Calogero, R. Pini. “c Horizontal Convexity on Carnot Groups.” Journal of Convex Analysis 19 (2012), No. 2, 541–567. https://doi.org/10.68381/jca19030

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press.