On Weakly H-Quasiconvex Functions on the Heisenberg Group

Andrea Calogero; Giovanna Carcano; Rita Pini

doi:10.68381/jca15050

Abstract

The aim of this paper is the investigation of weakly H-quasiconvex functions in the framework of the Heisenberg group

\mathbb{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

\mathbb{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.

Author information

Contact details are reproduced from the original publication and may be historical.

Andrea Calogero

Dip. di Statistica, Università 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, Università 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, Università di Milano-Bicocca, Piazza Ateneo Nuovo 1, 20126 Milano, Italy

rita.pini@unimib.it

Keywords

Heisenberg group
weak H-convexity
weak H-quasiconvexity
symmetrized horizontal Hessian

Mathematics Subject Classification

52A01; 53C17, 26B25

References

M. Avriel, W. E. Diewert, I. Zang: Nine kinds of quasiconcavity and concavity, J. Econ. Theory 25 (1981) 397–420.https://doi.org/10.1016/0022-0531(81)90039-9 Google Scholar
Z. M. Balogh, R. Monti: Accessible domains in the Heisenberg group, Proc. Amer. Math. Soc. 132 (2003) 97–106.https://doi.org/10.1090/S0002-9939-03-06978-8 Google Scholar
Z. M. Balogh, M. Rickly: Regularity of convex functions on Heisenberg groups, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 2 (2003) 847–868.Google Scholar
A. Bellaïche, J.-J. Risler (eds.): Sub–Riemannian Geometry, Progress in Mathematics 144, Birkhäuser, Basel (1996).https://doi.org/10.1007/978-3-0348-9210-0 Google Scholar
A. Calogero, G. Carcano, R. Pini: Optimization in the Heisenberg group, Optimization 55 (2006) 387–403.https://doi.org/10.1080/02331930600662880 Google Scholar
A. Calogero, G. Carcano, R. Pini: Twisted convex hulls in the Heisenberg group, J. Convex Analysis 14 (2007) 607–619.https://doi.org/10.68381/jca14036 Google Scholar
J. P. Crouzeix: On second order conditions for quasiconvexity, Math. Program. 18 (1980) 349–352.https://doi.org/10.1007/BF01588330 Google Scholar
D. Danielli, N. Garofalo, D. M. Nhieu: Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003) 263–341.https://doi.org/10.4310/CAG.2003.v11.n2.a5 Google Scholar
P. Juutinen, G. Lu, J. J. Manfredi, B. Stroffolini: Convex functions on Carnot groups, Rev. Mat. Iberoam. 23 (2007) 191–200.https://doi.org/10.4171/rmi/490 Google Scholar
G. Lu, J. J. Manfredi, B. Stroffolini: Convex functions on the Heisenberg group, Calc. Var. Partial Differ. Equ. 19 (2004) 1–22.https://doi.org/10.1007/s00526-003-0190-4 Google Scholar
V. Magnani: Lipschitz continuity, Aleksandrov theorem and characterizations for H–convex functions, Math. Ann. 334 (2006) 199–233.https://doi.org/10.1007/s00208-005-0717-4 Google Scholar
R. Monti: Some properties of Carnot–Carathéodory balls in the Heisenberg group, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 11 (2000) 155–167.Google Scholar
R. Monti, M. Rickly: Geodetically convex sets in the Heisenberg group, J. Convex Analysis 12 (2005) 187–196.https://doi.org/10.68381/jca12013 Google Scholar
M. B. Sun, X. P. Yang: Some properties of quasiconvex functions on the Heisenberg Group, Acta Math. Appl. Sin., Engl. Ser. 21(4) (2005) 571–580.https://doi.org/10.1007/s10255-005-0266-3 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.

[1] M. Avriel, W. E. Diewert, I. Zang: Nine kinds of quasiconcavity and concavity, J. Econ. Theory 25 (1981) 397–420.https://doi.org/10.1016/0022-0531(81)90039-9 Google Scholar

[2] Z. M. Balogh, R. Monti: Accessible domains in the Heisenberg group, Proc. Amer. Math. Soc. 132 (2003) 97–106.https://doi.org/10.1090/S0002-9939-03-06978-8 Google Scholar

[3] Z. M. Balogh, M. Rickly: Regularity of convex functions on Heisenberg groups, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 2 (2003) 847–868.Google Scholar

[4] A. Bellaïche, J.-J. Risler (eds.): Sub–Riemannian Geometry, Progress in Mathematics 144, Birkhäuser, Basel (1996).https://doi.org/10.1007/978-3-0348-9210-0 Google Scholar

[5] A. Calogero, G. Carcano, R. Pini: Optimization in the Heisenberg group, Optimization 55 (2006) 387–403.https://doi.org/10.1080/02331930600662880 Google Scholar

[6] A. Calogero, G. Carcano, R. Pini: Twisted convex hulls in the Heisenberg group, J. Convex Analysis 14 (2007) 607–619.https://doi.org/10.68381/jca14036 Google Scholar

[7] J. P. Crouzeix: On second order conditions for quasiconvexity, Math. Program. 18 (1980) 349–352.https://doi.org/10.1007/BF01588330 Google Scholar

[8] D. Danielli, N. Garofalo, D. M. Nhieu: Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003) 263–341.https://doi.org/10.4310/CAG.2003.v11.n2.a5 Google Scholar

[9] P. Juutinen, G. Lu, J. J. Manfredi, B. Stroffolini: Convex functions on Carnot groups, Rev. Mat. Iberoam. 23 (2007) 191–200.https://doi.org/10.4171/rmi/490 Google Scholar

[10] G. Lu, J. J. Manfredi, B. Stroffolini: Convex functions on the Heisenberg group, Calc. Var. Partial Differ. Equ. 19 (2004) 1–22.https://doi.org/10.1007/s00526-003-0190-4 Google Scholar

[11] V. Magnani: Lipschitz continuity, Aleksandrov theorem and characterizations for H–convex functions, Math. Ann. 334 (2006) 199–233.https://doi.org/10.1007/s00208-005-0717-4 Google Scholar

[12] R. Monti: Some properties of Carnot–Carathéodory balls in the Heisenberg group, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 11 (2000) 155–167.Google Scholar

[13] R. Monti, M. Rickly: Geodetically convex sets in the Heisenberg group, J. Convex Analysis 12 (2005) 187–196.https://doi.org/10.68381/jca12013 Google Scholar

[14] M. B. Sun, X. P. Yang: Some properties of quasiconvex functions on the Heisenberg Group, Acta Math. Appl. Sin., Engl. Ser. 21(4) (2005) 571–580.https://doi.org/10.1007/s10255-005-0266-3 Google Scholar