c Horizontal Convexity on Carnot Groups

Given a real-valued function

defined on the cartesian product of a generic Carnot group

\mathbf{G} G

and the first layer

V_1 V 1

of its Lie algebra, we introduce a notion of

c c

horizontal convex (

c c

H-convex) function on

\mathbf{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

c c

H-convex functions that, under appropriate assumptions on

c, c ,

are characterized by the nonemptiness of the

c 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 c

H-cyclic monotonicity for maps. In the particular case where

c(g,v)=\langle \xi_1(g),v \rangle, c ( g , v ) = ⟨ ξ 1 ( g ) , 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.