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Journal of Convex Analysis 15 (2008), No. 4, 753--766
Copyright Heldermann Verlag 2008



On Weakly H-Quasiconvex Functions on the Heisenberg Group

Andrea Calogero
Dip. di Statistica, UniversitÓ di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
andrea.calogero@unimib.it

Giovanna Carcano
Dip. di Metodi Quantitativi per le Scienze Economiche ed Aziendali, UniversitÓ di Milano-Bicocca, Piazza Ateneo Nuovo 1, 20126 Milano, Italy
giovanna.carcano@unimib.it

Rita Pini
Dip. di Metodi Quantitativi per le Scienze Economiche ed Aziendali, UniversitÓ di Milano-Bicocca, Piazza Ateneo Nuovo 1, 20126 Milano, Italy
rita.pini@unimib.it



The aim of this paper is the investigation of weakly H-quasiconvex functions in the framework of the Heisenberg group H. The functions of this class, recently introduced by M. B. Sun and X. P. Yang ["Some properties of quasiconvex functions on the Heisenberg Group", Acta Math. Appl. Sinica 21 (2005) 571--580], are defined via the property that their sublevels are weakly H-convex subsets of H. Following Crouzeix's approach, we prove conditions of first and second order, easily testable, for regular weakly H-quasiconvex functions, and we provide a characterization of them.

Keywords: Heisenberg group, weak H-convexity, weak H-quasiconvexity, symmetrized horizontal Hessian.

MSC: 52A01; 53C17, 26B25

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