Convex functions on the Heisenberg group

Abstract

Abstract. Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial differential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group.

A new characterization of convexity in free Carnot groups

“…The aim of this paper is to establish an analogue on Carnot groups G for vconvex functions, i.e., for functions which are convex in the viscosity sense first introduced by Lu, Manfredi and Stroffolini in [22] and by Juutinen, Lu, Manfredi and Stroffolini in [21]. Roughly speaking, these authors call v-convex any function which is subharmonic with respect to every sub-Laplacian of the stratified group G. On the other hand, if G is free, we know that every sub-Laplacian can be reduced to a single one via a "linear" change of variables; see [6].…”

“…Due to the relevance of the notion of convexity in theoretical and applied areas of mathematics, several notions of convexity have been recently proposed in the context of Carnot groups. The papers by Danielli, Garofalo and Nhieu [10] and by Lu, Manfredi and Stroffolini [22] opened the way for appropriate definitions of convexity in this context: they respectively introduced the classes of h-convex (horizontally convex) and v-convex (viscosity convex) functions. On the Heisenberg groups, these notions coincide (Balogh and Rickly [1]), whereas, generally, the v-convex functions are the u.s.c.…”

Convex analysis in groups and semigroups: a sampler

“…See also Remark 1 in [LMS04]. In a similar fashion to (1.2), one can define convex functions on additive groups and semigroups (again, see Section 2).…”

“…We refer the reader to [Mur03,vdV93] for more information on abstract convexity in all its manifestations. Some aspects of convex analysis in a more abstract setting have also been studied in [Ham05,JLMS07,LMS04]. Note that in [LMS04] for example, it is only required that a function is convex over geodesic curves (in this case, in the Heisenberg group).…”

“…Some aspects of convex analysis in a more abstract setting have also been studied in [Ham05,JLMS07,LMS04]. Note that in [LMS04] for example, it is only required that a function is convex over geodesic curves (in this case, in the Heisenberg group). Thus, the various notions of convexity do not always coincide.…”