Abstract

We study properties of functions convex with respect to a given family X\mathcal{X} of vector fields, a notion that appears natural in Carnot-Carathéodory metric spaces. We define a suitable subdifferential and show that a continuous function is X\mathcal{X}-convex if and only if such subdifferential is nonempty at every point. For vector fields of Carnot type we deduce from this property that a generalized Fenchel transform is involutive and a weak form of Jensen inequality. Finally we introduce and compare several notions of X\mathcal{X}-affine functions and show their connections with X\mathcal{X}-convexity.

Author information

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

Martino Bardi

Dip. di Matematica, Università di Padova, via Trieste 63, 35121 Padova, Italy

bardi@math.unipd.it

Federica Dragoni

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 2AG Wales, England

DragoniF@cardiff.ac.uk

Keywords

  • Convex functions in Carnot groups
  • Carnot-Carathéodory metric spaces
  • subdifferential
  • Legendre-Fenchel transform
  • convex duality
  • Jensen inequality

References

  1. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, New York (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  2. Z. Balogh, M. Rickly: Regularity of convex functions on Heisenberg groups, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 2(4) (2003) 847–868.Google Scholar
  3. M. Bardi, I. Capuzzo Dolcetta: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston (1997).https://doi.org/10.1007/978-0-8176-4755-1Google Scholar
  4. M. Bardi, F. Dragoni: Convexity and semiconvexity along vector fields, Calc. Var. Partial Differ. Equ. 42 (2011) 405–427.https://doi.org/10.1007/s00526-011-0392-0Google Scholar
  5. M. Bardi, P. Mannucci: Comparison principles for subelliptic equations of Monge-Ampère type, Boll. Unione Mat. Ital. (9) 1(2) (2008) 489–495.Google Scholar
  6. M. Bardi, P. Mannucci: Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge-Ampère type, Forum Math. 25 (2013) 1291–1330.https://doi.org/10.1515/forum-2013-0067Google Scholar
  7. A. Bellaïche, J.-J. Risler (eds.): Sub-Riemannian Geometry (Paris, 1992), Progress in Mathematics 144, Birkhäuser, Basel (1996).https://doi.org/10.1007/978-3-0348-9210-0Google Scholar
  8. A. Bonfiglioli, E. Lanconelli, F. Uguzzoni: Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Springer, Berlin (2007).Google Scholar
  9. A. Calogero, R. Pini: Note on the Fenchel transform in the Heisenberg group, J. Math. Anal. Appl. 368 (2010) 69–79.https://doi.org/10.1016/j.jmaa.2010.03.001Google Scholar
  10. A. Calogero, R. Pini: Horizontal normal map on the Heisenberg group, J. Nonlinear Convex Anal. 12 (2011) 287–307.Google Scholar
  11. L. Capogna, S. D. Pauls, J. T. Tyson: Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups, Trans. Amer. Math. Soc. 362 (2010) 4045–4062.https://doi.org/10.1090/S0002-9947-10-04768-9Google Scholar
  12. G. Citti, M. Manfredini, A. Sarti: From neural oscillations to variational problems in the visual cortex, J. Physiology 97 (2003) 87–385.https://doi.org/10.1016/j.jphysparis.2003.09.014Google Scholar
  13. G. Citti, M. Manfredini, A. Sarti: Neuronal oscillations in the visual cortex: Gamma-convergence to the Riemannian Mumford-Shah functional, SIAM J. Math. Anal. 35 (2004) 1394–1419.https://doi.org/10.1137/S0036141002398673Google Scholar
  14. G. Citti, A. Sarti: A cortical based model of perceptual completion in the rototraslation space, J. Math. Imaging Vis. 24 (2006) 307–326.https://doi.org/10.1007/s10851-005-3630-2Google Scholar
  15. D. Danielli, N. Garofalo, D. M. Nhieu: Notions of convexity in Carnot groups, Commun. Anal. Geom. 11 (2003) 263–341.https://doi.org/10.4310/CAG.2003.v11.n2.a5Google Scholar
  16. N. Garofalo, F. Tournier: New properties of convex functions in the Heisenberg group, Trans. Amer. Math. Soc. 358(5) (2006) 2011–2055.https://doi.org/10.1090/S0002-9947-05-04016-XGoogle Scholar
  17. C. E. Gutiérrez, A. Montanari: Maximum and comparison principles for convex functions on the Heisenberg group, Commun. Partial Differ. Equations 29(9–10) (2004) 1305–1334.https://doi.org/10.1081/PDE-200037752Google Scholar
  18. C. E. Gutiérrez, A. Montanari: On the second order derivatives of convex functions on the Heisenberg group, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 3(2) (2004) 349–366.https://doi.org/10.2422/2036-2145.2004.2.03Google Scholar
  19. P. Juutinen, G. Lu, J. J. Manfredi, B. Stroffolini: Convex functions on Carnot groups, Rev. Mat. Iberoam. 23(1) (2007) 191–200.https://doi.org/10.4171/rmi/490Google Scholar
  20. G. Lu, J. J. Manfredi, B. Stroffolini: Convex functions in the Heisenberg group, Calc. Var. Partial Differ. Equ. 19(1) (2004) 1–22.https://doi.org/10.1007/s00526-003-0190-4Google Scholar
  21. V. Magnani: Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions, Math. Ann. 334(1) (2006) 199–233.https://doi.org/10.1007/s00208-005-0717-4Google Scholar
  22. V. Magnani, M. Scienza: Characterization of differentiability for h-convex functions in stratified groups, Ann. Sc. Norm. Super. Pisa, Cl. Sci., to appear.Google Scholar
  23. V. Magnani, M. Scienza: Regularity estimates for convex functions in Carnot-Carathéodory spaces, preprint (2012).Google Scholar
  24. R. Montgomery: A Tour of Subriemannian Geometries, their Geodesics and Applications, Mathematical Surveys and Monographs 91, American Mathematical Society, Providence (2002).Google Scholar
  25. R. Monti, M. Rickly: Geodetically convex sets in the Heisenberg group, J. Convex Analysis 12(1) (2005) 187–196.https://doi.org/10.68381/jca12013Google Scholar
  26. C. Wang: Viscosity convex functions on Carnot groups, Proc. Amer. Math. Soc. 133(4) (2005) 1247–1253.https://doi.org/10.1090/S0002-9939-04-07836-0Google Scholar

Suggested citation

M. Bardi, F. Dragoni. “Subdifferential and Properties of Convex Functions with Respect to Vector Fields.” Journal of Convex Analysis 21 (2014), No. 3, 785–810. https://doi.org/10.68381/jca21042

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.