Abstract

We prove that the geodetic envelope of a subset of the Heisenberg group containing three points not lying on the same geodesic is the whole group. As a corollary, we obtain that a function on the group which is convex along geodesics must be constant

Author information

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

Roberto Monti

Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Permanent Address: Università di Padova, Dipartimento di Matematica Pura e Applicata, Via Belzoni 7, 35131 Padova, Italy

monti@math.unipd.it

References

  1. L. Ambrosio, S. Rigot: Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 203 (2004) 261–301.https://doi.org/10.1016/S0022-1236(03)00019-3Google Scholar
  2. A. Bellaïche: The tangent space in sub-Riemannian geometry, in: Sub-Riemannian Geometry, Prog. Math. 144, Birkhäuser, Basel (1996) 1–78.https://doi.org/10.1007/978-3-0348-9210-0_1Google Scholar
  3. Z. M. Balogh, M. Rickly: Regularity of convex functions on Heisenberg groups, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (5) Vol. II (2003) 847–868.Google Scholar
  4. 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
  5. B. Gaveau: Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977) 95–153.https://doi.org/10.1007/BF02392235Google Scholar
  6. C. E. Gutiérrez, A. Montanari: Maximum and comparison principles for convex functions on the Heisenberg group, Commun. Partial Differ. Equations, to appear.Google Scholar
  7. N. Garofalo, F. Tournier: New properties of convex functions in the Heisenberg group, Trans. Am. Math. Soc., to appear.Google Scholar
  8. A. Korányi: Geometric properties of Heisenberg–type groups, Adv. Math. 56 (1985) 28–38.https://doi.org/10.1016/0001-8708(85)90083-0Google Scholar
  9. G. Lu, J. Manfredi, B. Stroffolini: Convex functions on the Heisenberg group, Calc. Var. Partial Differ. Equ. 19 (2003) 1–22.https://doi.org/10.1007/s00526-003-0190-4Google Scholar
  10. R. Monti: Some properties of Carnot-Carathéodory balls in the Heisenberg group, Rend. Mat. Acc. Lincei, s. 9, v. 11 (2000) 155–167.Google Scholar

Suggested citation

R. Monti, M. Rickly. “Geodetically Convex Sets in the Heisenberg Group.” Journal of Convex Analysis 12 (2005), No. 1, 187–196. https://doi.org/10.68381/jca12013

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2005. Rights now held by Banach Press LLC.