A Sard theorem for tame set-valued mappings

Journal of Mathematical Analysis and Applications

Volumn 335, Issue 2, 2007, Pages 882-901

  Ioffe, A ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Critical value; Definable set valued mapping; o Minimal structure; Rate of surjection

Indexed keywords

EID: 34447324163     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2007.01.104     Document Type: Article

References (28)

1. d'Acunto D. Valeurs critiques asymptotiques de fonctions définissables dans un structure o-minimale. Ann. Polon. Math. LXXV (2000) 35-45
2. Azé D., and Corvellec J.-N. Characterization of error bounds for lower semicontinuous functions on metric spaces. ESAIM Control Optim. Calc. Var. 10 (2004) 409-425
3. Bates S.M. Toward a precise smoothness hypothesis in Sard's theorem. Proc. Amer. Math. Soc. 117 (1993) 279-283
4. Bolte J., Daniilidis A., and Lewis A. A nonsmooth Morse-Sard theorem for subanalytic functions. J. Math. Anal. Appl. 321 2 (2006) 729-740
5. J. Bolte, A. Daniilidis, A. Lewis, The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim., in press
6. J. Bolte, A. Daniilidis, A. Lewis, M. Shiota, A Sard-type theorem for Clarke critical values of subanalytic Lipschitz continuous functions, Ann. Polon. Math., in press
7. Bochnak J., Coste M., and Roy M.-F. Real Algebraic Geometry (1998), Springer
8. Borwein J.M., and Zhu Q.J. Techniques of Variational Analysis (2005), Springer
9. Coste M. An introduction to o-minimal geometry. Inst. Rech. Math., Univ. de Rennes (1999). http://name.math.univ-rennes1.fr/michel.coste/polyens/OMIN.pdf
10. De Giorgi E., Marino A., and Tosques M. Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. 68 (1980) 180-187
11. van den Dries L. Tame Topology and o-Minimal Structures (1998), Cambridge Univ. Press
12. Grothendieck A. Sketch of a proposal. In: Schneps L., and Lochak P. (Eds). Geometric Galois Actions (1997), Cambridge Univ. Press
13. Ioffe A.D. Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55 3 (2000) 103-162 English translation:
14. Ioffe A.D. Metric regularity and subdifferential calculus. Russian Math. Surveys 55 (2000) 501-558
15. Kurdyka K., Orro P., and Simon S. Semialgebraic Sard theorem for generalized critical values. J. Differential Geom. 56 (2000) 67-92
16. Lewis A.S. Active sets, nonsmoothness and sensitivity. SIAM J. Optim. 13 (2003) 702-725
17. Mifflin R. Semismooth and semiconvex functions in constrained optimization. Math. Oper. Res. 2 (1977) 191-207
18. Mordukhovich B.S. Variational Analysis and Generalized Differentiation, vol. 1 (2005), Springer
19. Norton A. Functions not constant on fractal quasi-arcs of critical points. Proc. Amer. Math. Soc. 106 (1989) 397-405
20. Rifford L. A Morse-Sard theorem for the distance function on Riemannian manifolds. Manuscripta Math. 113 (2004) 251-265
21. Rockafellar R.T. Favorable classes of Lipschitz continuous functions in subgradient optimization. In: Nurminski E. (Ed). Progress in Non-Differentiable Optimization (1981), Pergamon Press
22. Rockafellar R.T., and Wets R.J.B. Variational Analysis (1998), Springer
23. Rohde A. On Sard's theorem for nonsmooth functions. Numer. Funct. Anal. Optim. 9-10 (1997) 1023-1039
24. Teissier B. Tame and stratified objects. In: Schneps L., and Lochak P. (Eds). Geometric Galois Actions (1997), Cambridge Univ. Press 231-242
25. Whitney H. A function not constant on a connected set of critical points. Duke Math. J. 1 (1935) 514-517
26. Y. Yomdin, Maxima of smooth families III: Morse-Sard theorem, preprint, MOI, Bonn, 1984
27. Yomdin Y., and Comte G. Tame Geometry with Applications to Smooth Analysis. Lecture Notes in Math. vol. 1834 (2004), Springer
28. Yomdin Y. Surjective mapping whose differential is nowhere surjective. Proc. Amer. Math. Soc. 111 (1991) 267-270