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Quasi-convex functions in Carnot groups. (English) Zbl 1123.43004
The authors introduce the concept of $$h$$-quasiconvexity which generalizes the notion of $$h$$-convexity in the Carnot group $$G$$. An example of $$h$$-quasiconvex function which is not $$h$$-convex is provided. Some interesting properties similar to those of $$h$$-convex functions on $$G$$ are given. In particular, the authors show that the notions of $$h$$-quasiconvex functions and $$h$$-convex sets are equivalent, give the $$L^\infty$$ estimates of the first derivatives of $$h$$-quasiconvex functions, and prove that $$h$$-quasiconvex functions on a Carnot group $$G$$ of step two are locally bounded from above. Furthermore, for a Carnot group $$G$$ of step two, the authors obtain that $$h$$-convex functions are locally Lipschitz continuous and twice differentiable almost everywhere in $$G$$. This last result is the version of the Busemann-Feller-Alexandrov theorem for the class of $$h$$-convex functions in Carnot groups of step two.

##### MSC:
 43A80 Analysis on other specific Lie groups 43-06 Proceedings, conferences, collections, etc. pertaining to abstract harmonic analysis 26B25 Convexity of real functions of several variables, generalizations
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