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On the metric projection onto prox-regular subsets of Riemannian manifolds


Authors: Seyedehsomayeh Hosseini and Mohamad R. Pouryayevali
Journal: Proc. Amer. Math. Soc. 141 (2013), 233-244
MSC (2010): Primary 49J52, 58C06, 58C20
DOI: https://doi.org/10.1090/S0002-9939-2012-11828-3
Published electronically: September 10, 2012
MathSciNet review: 2988725
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Abstract: Prox-regular subsets of Riemannian manifolds are introduced. A characterization of prox-regular sets based on the hypomonotonicity of the truncated limiting normal cone is obtained. Moreover, some properties of metric projection mapping and distance function corresponding to the prox-regular sets are presented.


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  • 1. D. Azagra, J. Ferrera, F. López-Mesas, Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds, J. Funct. Anal. 220 (2005) 304-361. MR 2119282 (2005k:49045)
  • 2. D. Azagra, J. Ferrera, Proximal calculus on Riemannian manifolds, Mediterr. J. Math. 2 (2005) 437-450. MR 2192524 (2007a:49023)
  • 3. A. Barani, M. R. Pouryayevali, Invariant monotone vector fields on Riemannian manifolds, Nonlinear Anal. 70 (2009) 1850-1861. MR 2492123 (2010e:46038)
  • 4. R. D. Canary, D. B. A. Epstein, A. Marden, Fundamentals of Hyperbolic Geometry: Selected Expositions, Cambridge University Press, 2006. MR 2230672 (2007c:57002)
  • 5. F. H. Clarke, Yu. S. Ledayaev, R. J. Stern, P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, Vol. 178, Springer, New York, 1998. MR 1488695 (99a:49001)
  • 6. M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992. MR 1138207 (92i:53001)
  • 7. J. X. Da Cruz Neto, O. P. Ferreira, L. R. Lucambio Pérez, Contributions to the study of monotone vector fields, Acta Math. Hungar. 94 (2002) 307-320. MR 1905099 (2003e:58023)
  • 8. R. E. Greene, K. Shiohama, Convex functions on complete noncompact manifolds: topological structure, Invent. Math. 63 (1981) 129-157. MR 608531 (82e:53065)
  • 9. S. Grognet, Théorème de Motzkin en courbure négative, Geom. Dedicata 79 (2000) 219-227. MR 1748883 (2001a:53066)
  • 10. S. Hosseini, M. R. Pouryayevali, Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds, Nonlinear Anal. 74 (2011) 3884-3895. MR 2802974
  • 11. W. Klingenberg, Riemannian Geometry, Walter de Gruyter Studies in Mathematics, Vol. 1, Walter de Gruyter, Berlin, New York, 1982. MR 666697 (84j:53001)
  • 12. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. II, Wiley-Interscience, New York, 1969. MR 0238225 (38:6501)
  • 13. S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Vol. 191, Springer, New York, 1999. MR 1666820 (99m:53001)
  • 14. J. M. Lee, Riemannian Manifolds, An Introduction to Curvature, Graduate Texts in Mathematics, Vol. 176, Springer, 1997. MR 1468735 (98d:53001)
  • 15. D. Motreanu, N. H. Pavel, Quasitangent vectors in flow-invariance and optimization problems on Banach manifolds. J. Math. Anal. Appl. 88 (1982) 116-132. MR 661406 (83h:58086)
  • 16. S. Z. Németh, Monotone vector fields, Publ. Math. Debrecen 54 (1999) 437-449. MR 1694468 (2000g:53041)
  • 17. R. A. Poliquin, R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc. 348 (1996) 1805-1838. MR 1333397 (96h:49039)
  • 18. R. A. Poliquin, R. T. Rockafellar, L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000) 5231-5249. MR 1694378 (2001b:49024)
  • 19. A. Shapiro, Existence and differentiability of metric projections in Hilbert spaces. SIAM J. Optim. 4 (1994) 130-141. MR 1260410 (94m:90111)
  • 20. A. Shapiro, Perturbation analysis of optimization problems in Banach spaces, Numer. Funct. Anal. Optim. 13 (1992) 97-116. MR 1163320 (93c:49019)
  • 21. R. Walter, On the metric projection onto convex sets in Riemannian spaces, Arch. Math. (Basel) 25 (1974) 91-98. MR 0397631 (53:1490)
  • 22. J. Wang, G. López, V. Martín-Márquez, C. Li, Monotone and accretive vector fields on Riemannian manifolds, J. Optim. Theory Appl. 146 (2010) 691-708. MR 2720608 (2012b:47217)
  • 23. J. H. C. Whitehead, Convex regions in the geometry of paths. Q. J. Math. 3 (1932) 33-42.

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Additional Information

Seyedehsomayeh Hosseini
Affiliation: Department of Mathematics, University of Isfahan, P. O. Box 81745-163, Isfahan, Iran
Email: somayeh-hosseini@hotmail.com

Mohamad R. Pouryayevali
Affiliation: Department of Mathematics, University of Isfahan, P. O. Box 81745-163, Isfahan, Iran
Email: pourya@math.ui.ac.ir

DOI: https://doi.org/10.1090/S0002-9939-2012-11828-3
Keywords: Prox-regularity, Riemannian manifolds, Clarke subdifferential, proximal subdifferential.
Received by editor(s): December 18, 2010
Published electronically: September 10, 2012
Additional Notes: The second author was partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran.
Communicated by: Sergei K. Suslov
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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