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) is the product of the projection of g on the first layer with v, 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.

Calogero, A., Pini, R. (2012). c Horizontal Convexity on Carnot Groups. JOURNAL OF CONVEX ANALYSIS, 19(2), 541-567.

c Horizontal Convexity on Carnot Groups

CALOGERO, ANDREA GIOVANNI
;
PINI, RITA
2012

Abstract

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) is the product of the projection of g on the first layer with v, 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.
Articolo in rivista - Articolo scientifico
Carnot group, horizontal convexity, c horizontal convexity, c horizontal differential, c horizontal cyclic monotonicity.
English
2012
19
2
541
567
none
Calogero, A., Pini, R. (2012). c Horizontal Convexity on Carnot Groups. JOURNAL OF CONVEX ANALYSIS, 19(2), 541-567.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/30644
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