Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T01:00:07.547Z Has data issue: false hasContentIssue false
  • English
  • Français

Proximal Analysis and Boundaries of Closed Sets in Banach Space. Part II: Applications

Published online by Cambridge University Press:  20 November 2018

J. M. Borwein  and
H. M. Strojwas
J. M. Borwein
Affiliation:
Dalhousie University, Halifax, Nova Scotia
H. M. Strojwas
Affiliation:
Carnegie-Mellon University, Pittsburgh, Pennsylvania
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is a direct continuation of the article “Proximal analysis and boundaries of closed sets in Banach space, Part I: Theory”, by the same authors. It is devoted to a detailed analysis of applications of the theory presented in the first part and of its limitations.

Theorem 2.1 has important consequences for geometry of Banach spaces. We start the presentation with a discussion of density and existence of R-proper points (Definition 1.3) for closed sets in Banach spaces. Our considerations will be based on the “lim inf” inclusions proven in the first part of our paper.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 2 , 01 April 1987 , pp. 428 - 472
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Asplund, E., Frechet-differentiability of convex functions, Acta Math. 121 (1968), 31–47.Google Scholar
2. Bishop, E. and Phelps, R. R., The support functionals of a convex set, Proc. Symp. Amer. Math. Soc. 139 (1963), 433–467.Google Scholar
3. Borwein, J. M., Tangent cones, starshape arid convexity, Internat. J. Math. & Math. Sci. 1 (1978), 459–477.Google Scholar
4. Borwein, J. M. and O'Brien, R., Tangent cones and convexity, Canadian Math. Bull. 19 (1976), 257–261.Google Scholar
5. Borwein, J. M., Edelstein, M. and O'Brien, R., Visibility and starshape, J. London Math. Soc. (2), 4 (1976), 313–318.Google Scholar
6. Borwein, J. M. and Strojwas, H. M., Tangential approximations, to appear in Nonlinear Anal. Th. Meth. Appl. 9 (1985), 1347–1366.Google Scholar
7. Borwein, J. M. and Strojwas, H. M., Proximal analysis and boundaries of closed sets in Banach space. Part I: Theory, Can. J. Math. 38 (1986), 431–452.Google Scholar
8. Borwein, J. M. and Tingley, D. W., On supportless convex sets, Proc. Amer. Math. Soc. 04 (1985), 471–476.Google Scholar
9. Clarke, F. H., Optimization and nonsmooth analysis (John Wiley, 1982).Google Scholar
10. Clarke, F. H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262.Google Scholar
11. Diestel, J., Geometry of Banach spaces-selected topics (Springer-Verlag, 1975).CrossRefGoogle Scholar
12. Holmes, R. B., Geometric functional analysis and its applications (Springer-Verlag, 1975).CrossRefGoogle Scholar
13. Krasnoselski, M., Sur un critère pour qu'un domaine soit étaillée, Rec. Math. [Math. Sbornik] N. S. 19 (1946), 309–310.Google Scholar
14. Larman, D. G. and Phelps, R., Gateaux differentiability of convex functions on Banach spaces, J. London Math. Soc. (2), 20 (1979), 115–127.Google Scholar
15. Lebourg, G., Generic differentiability of Lipschitzian functions, Trans. Am. Math. Soc. 256 (1979), 125–144.Google Scholar
16. McLinden, M., An application of Ekeland's theorem to mimimax problems, Nonlinear. Anal. Th. Meth. Appl. 6 (1982), 189–196.Google Scholar
17. Penot, J. P., A characterization of tangential regularity, Nonlinear Anal., Theory, Meth. Appl. 5 (1981), 625–633.Google Scholar
18. Rockafellar, R. T., Generalized directional derivatives and subgradients of nonconvex functions, Can. J. Math. 32 (1980), 257–280.Google Scholar
19. Rockafellar, R. T., Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Mathematics of Operations Research 6 (1981).Google Scholar
20. Treiman, J. S., Characterization of Clarke's tangent and normal cones infinite and infinite dimensions, Nonlinear Anal. Th. Meth. Appl. 7 (1983), 771–783.Google Scholar
21. Yamamuro, S., Differential calculus in topological linear spaces (Springer-Verlag, 1970).Google Scholar
22. Zhivkov, N. V., Generic Gateaux differentiability of directionally differentiable mappings, (submitted to Rev. Roum. Math. Pures Appl., 1984).Google Scholar