Mathematics – Differential Geometry
Scientific paper
2010-05-06
Mathematics
Differential Geometry
Scientific paper
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 $c(g,v)=< \xi_1(g),v >,$ 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.
Calogero Andrea
Pini Rita
No associations
LandOfFree
c horizontal convexity on Carnot groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with c horizontal convexity on Carnot groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and c horizontal convexity on Carnot groups will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-26306