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

Borwein, J. M. ; Strojwas, H. M.

Canadian journal of mathematics, Tome 39 (1987) no. 2, pp. 428-472

Résumé

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.

Détail

DOI : 10.4153/CJM-1987-019-4

Bibliographie

[1] 1. Asplund, E., Frechet-differentiability of convex functions, Acta Math. 121 (1968), 31–47. Google Scholar

[2] 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] 3. Borwein, J. M., Tangent cones, starshape arid convexity, Internat. J. Math. & Math. Sci. 1 (1978), 459–477. Google Scholar

[4] 4. Borwein, J. M. and O'Brien, R., Tangent cones and convexity, Canadian Math. Bull. 19 (1976), 257–261. Google Scholar

[5] 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] 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] 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] 8. Borwein, J. M. and Tingley, D. W., On supportless convex sets, Proc. Amer. Math. Soc. 04 (1985), 471–476. Google Scholar

[9] 9. Clarke, F. H., Optimization and nonsmooth analysis (John Wiley, 1982). Google Scholar

[10] 10. Clarke, F. H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262. Google Scholar

[11] 11. Diestel, J., Geometry of Banach spaces-selected topics (Springer-Verlag, 1975). Google Scholar | DOI

[12] 12. Holmes, R. B., Geometric functional analysis and its applications (Springer-Verlag, 1975). Google Scholar | DOI

[13] 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] 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] 15. Lebourg, G., Generic differentiability of Lipschitzian functions, Trans. Am. Math. Soc. 256 (1979), 125–144. Google Scholar

[16] 16. McLinden, M., An application of Ekeland's theorem to mimimax problems, Nonlinear. Anal. Th. Meth. Appl. 6 (1982), 189–196. Google Scholar

[17] 17. Penot, J. P., A characterization of tangential regularity, Nonlinear Anal., Theory, Meth. Appl. 5 (1981), 625–633. Google Scholar

[18] 18. Rockafellar, R. T., Generalized directional derivatives and subgradients of nonconvex functions, Can. J. Math. 32 (1980), 257–280. Google Scholar

[19] 19. Rockafellar, R. T., Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Mathematics of Operations Research 6 (1981). Google Scholar

[20] 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] 21. Yamamuro, S., Differential calculus in topological linear spaces (Springer-Verlag, 1970). Google Scholar

[22] 22. Zhivkov, N. V., Generic Gateaux differentiability of directionally differentiable mappings, (submitted to Rev. Roum. Math. Pures Appl., 1984). Google Scholar