Journal of Convex Analysis 19 (2012), No. 2, 541--567
Copyright Heldermann Verlag 2012
c Horizontal Convexity on Carnot Groups
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
[Abstract-pdf]
\def\G{\mathbf{G}} Given a real-valued function $c$ defined on the cartesian product of a generic Carnot group $\G$ and the first layer $V_1$ of its Lie algebra, we introduce a notion of $c$ horizontal convex ($c$ H-convex) function on $\G$ 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 $c$ H-convex functions that, under appropriate assumptions on $c,$ are characterized by the nonemptiness of the $c$ 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 $c$ H-cyclic monotonicity for maps. In the particular case where $c(g,v)=\langle \xi_1(g),v \rangle,$ 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.
Keywords: Carnot group, horizontal convexity, c horizontal convexity, c horizontal differential, c horizontal cyclic monotonicity.
MSC: 52A01; 22E25
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