c Horizontal Convexity on Carnot Groups

\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.