Abstract

We present some properties of tangent and normal cones of epi-Lipschitz subsets of complete Riemannian manifolds. The fact that epi-Lipschitz subsets of complete Riemannian manifolds are absolute neighborhood retracts is proved. A notion of Euler characteristic of epi-Lipschitz subsets of complete Riemannian manifolds is introduced. Moreover, we provide a sufficient condition which ensures that the Euler characteristic of this class of sets is equal to one. Then, these results are applied to equilibrium theory on complete parallelizable Riemannian manifolds.

Author information

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

S. Hosseini

Dept. of Mathematics, University of Isfahan, P. O. Box 81745-163, Isfahan, Iran

hoseini@math.ui.ac.ir

Mohamad Reza Pouryayevali

Dept. of Mathematics, University of Isfahan, P. O. Box 81745-163, Isfahan, Iran

pourya@math.ui.ac.ir

Keywords

  • Clarke subdifferential
  • Epi-Lipschitz sets
  • Euler characteristic
  • Riemannian manifolds

Mathematics Subject Classification

49J52, 58E05, 58C30, 55M25

References

  1. A. A. Agrachev, D. Pallaschke, S. Scholtes: On Morse theory for piecewise smooth functions, J. Dyn. Control Syst. 3 (1997) 449–469.https://doi.org/10.1023/A:1021810206861Google Scholar
  2. D. Azagra, J. Ferrera, F. López-Mesas: Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds, J. Funct. Anal. 220 (2005) 304–361.https://doi.org/10.1016/j.jfa.2004.10.008Google Scholar
  3. D. Azagra, J. Ferrera: Applications of proximal calculus to fixed point theory on Riemannian manifolds, Nonlinear. Anal. 67 (2007) 154–174.https://doi.org/10.1016/j.na.2006.04.019Google Scholar
  4. H. Ben-El-Mechaiekh, W. Kryszewski: Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc. 349 (1997) 4159–4179.https://doi.org/10.1090/S0002-9947-97-01836-9Google Scholar
  5. J. M. Bonnisseau, B. Cornet: Fixed-point theorems and Morse’s lemma for Lipschitzian functions, J. Math. Anal. Appl. 146 (1990) 318–332.https://doi.org/10.1016/0022-247X(90)90305-YGoogle Scholar
  6. J. M. Bonnisseau, B. Cornet, M.-O. Czarnecki: The marginal pricing rule revisited, Econ. Theory 33 (2007) 579–589.https://doi.org/10.1007/s00199-006-0160-0Google Scholar
  7. W. M. Boothby: An introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed., Academic Press, Orlando (1986).Google Scholar
  8. J. M. Borwein, Y. Lucet, B. Mordukhovich: Compactly epi-Lipschitzian convex sets and functions in normed spaces, J. Convex Analysis 7 (2000) 375–393.https://doi.org/10.68381/jca07019Google Scholar
  9. R. D. Canary, D. B. A. Epstein, A. Marden (eds.): Fundamentals of Hyperbolic Geometry: Selected Expositions, Cambridge University Press, Cambridge (2006).https://doi.org/10.1017/CBO9781139106986Google Scholar
  10. A. Cellina: A theorem on the approximation of compact multivalued mappings, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 47 (1969) 429–433.Google Scholar
  11. A. Cellina, A. Lasota: A new approach to the definition of topological degree for multi-valued mappings, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 46 (1969) 434–440.Google Scholar
  12. F. H. Clarke, Yu. S. Ledayaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics 178, Springer, New York (1998).Google Scholar
  13. B. Cornet, M.-O. Czarnecki: Smooth representations of epi-Lipschitzian subsets of Rn, Nonlinear Anal. 37 (1999) 139–160.https://doi.org/10.1016/S0362-546X(98)00033-9Google Scholar
  14. B. Cornet: Euler characteristic and fixed point theorems, Positivity 6 (2002) 243–260.https://doi.org/10.1023/A:1020242731195Google Scholar
  15. B. Cornet, M.-O. Czarnecki: Existence of (generalized) equilibria on an epi-Lipschitzian domain: A necessary and sufficient condition, Cahier Eco-Maths 95-55, Université de Paris 1, Paris (1995).Google Scholar
  16. A. Ćwiszewski, W. Kryszewski: Homotopy invariants for tangent vector fields on closed sets, Nonlinear. Anal. 65 (2006) 175–209.https://doi.org/10.1016/j.na.2005.09.010Google Scholar
  17. M.-O. Czarnecki, L. Rifford: Approximation and regularization of Lipschitz functions; convergence of the gradients, Trans. Amer. Math. Soc. 358 (2006) 4467–4520.https://doi.org/10.1090/S0002-9947-06-04103-1Google Scholar
  18. M. P. Do Carmo: Riemannian Geometry, Birkhäuser, Boston (1992).https://doi.org/10.1007/978-1-4757-2201-7Google Scholar
  19. R. E. Greene, K. Shiohama: Convex functions on complete noncompact manifolds: Topological structure, Invent. Math. 63 (1981) 129–157.https://doi.org/10.1007/BF01389196Google Scholar
  20. A. Gudovich, M. Kamenskii, M. Quincampoix: Existence of equilibria of set-valued maps on bounded epi-Lipschitz domains in Hilbert spaces without invariance conditions, Nonlinear Anal. 72 (2010) 262–276.https://doi.org/10.1016/j.na.2009.06.060Google Scholar
  21. M. W. Hirsch: Differential Topology, Graduate Texts in Mathematics 33, Springer, New York (1976).https://doi.org/10.1007/978-1-4684-9449-5Google Scholar
  22. S. Hosseini, M. R. Pouryayevali: Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds, Nonlinear Anal. 74 (2011) 3884–3895.https://doi.org/10.1016/j.na.2011.02.023Google Scholar
  23. V. Jeyakumar, D. T. Luc: Nonsmooth Vector Functions and Continuous Optimization, Springer, New York (2008).Google Scholar
  24. A. Jourani: On a class of compactly epi-Lipschitzian sets, Nonlinear Anal. 54 (2003) 471–483.https://doi.org/10.1016/S0362-546X(03)00106-8Google Scholar
  25. W. Klingenberg: Riemannian Geometry, De Gruyter Studies in Mathematics 1, Walter de Gruyter, Berlin (1982).Google Scholar
  26. A. Kristály: Economic Optimization Problems via Riemann-Finsler Geometry, PhD Thesis, Central European University, Budapest (2010).Google Scholar
  27. A. Kristály: Location of Nash equilibria: a Riemannian geometrical approach, Proc. Amer. Math. Soc. 138 (2010) 1803–1810.https://doi.org/10.1090/S0002-9939-09-10145-4Google Scholar
  28. S. Lang: Fundamentals of Differential Geometry, Graduate Texts in Mathematics 191, Springer, New York (1999).https://doi.org/10.1007/978-1-4612-0541-8Google Scholar
  29. Yu. S. Ledyaev, Q. J. Zhu: Nonsmooth analysis on smooth manifolds, Trans. Amer. Math. Soc. 359 (2007) 3687–3732.https://doi.org/10.1090/S0002-9947-07-04075-5Google Scholar
  30. J. M. Lee: Riemannian Manifolds. An Introduction to Curvature, Graduate Texts in Mathematics 176, Springer, New York (1997).https://doi.org/10.1007/b98852Google Scholar
  31. C. Li, B. S. Mordukhovich, J. Wang, J. C. Yao: Weak sharp minima on Riemannian manifolds, SIAM J. Optim. 21 (2011) 1523–1560.https://doi.org/10.1137/09075367XGoogle Scholar
  32. T. Lorenz: Epi-Lipschitzian reachable sets of differential inclusions, Syst. Control Lett. 57 (2008) 703–707.https://doi.org/10.1016/j.sysconle.2008.01.007Google Scholar
  33. J. Milnor: Topology from the Differentiable Viewpoint, University Press of Virginia, Charlottesville (1965).Google Scholar
  34. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. I: Basic Theory, Grundlehren der Mathematischen Wissenschaften 331, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  35. R. T. Rockafellar: Clarke’s tangent cones and the boundaries of closed sets in Rn, Nonlinear Anal. 3 (1979) 145–154.https://doi.org/10.1016/0362-546X(79)90044-0Google Scholar
  36. R. T. Rockafellar: Directionally Lipschitzian functions and subdifferential calculus, Proc. Lond. Math. Soc. 39 (1979) 331–355.https://doi.org/10.1112/plms/s3-39.2.331Google Scholar
  37. J. T. Schwartz: Nonlinear Functional Analysis, Gordon and Breach, New York (1969).Google Scholar

Suggested citation

S. Hosseini, M. R. Pouryayevali. “Euler Characteristic of Epi-Lipschitz Subsets of Riemannian Manifolds.” Journal of Convex Analysis 20 (2013), No. 1, 67–91. https://doi.org/10.68381/jca20005

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press.