Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T00:33:31.054Z Has data issue: false hasContentIssue false
  • English
  • Français

Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space

Published online by Cambridge University Press:  20 November 2018

R. R. Phelps
R. R. Phelps*
Affiliation:
Department of Mathematics GN-50, University of Washington, SeattleWA 98195
Rights & Permissions [Opens in a new window]

Abstract

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.

The Bishop-Phelps theorem guarantees the existence of support points and support functionals for a nonempty closed convex subset of a Banach space; equivalently, it guarantees the existence of subdifferentials and points of subdifferentiability of a proper lower semicontinuous convex function on a Banach space. In this note we show that most of these results cannot be extended to pairs of convex sets or functions, even in Hilbert space. For instance, two proper lower semicontinuous convex functions need not have a common point of subdifferentiability nor need they have a subdifferential in common. Negative answers are also obtained to certain questions concerning density of support points for the closed sum of two convex subsets of Hilbert space.

Keywords

46B2046B2246C0552A05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 1 , 01 March 1988 , pp. 121 - 128
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Borwein, J. M., Continuity and differentiability properties of convex operators, Proc. London Math. Soc. 3 (44) (1982), pp. 420–444.Google Scholar
2. Borwein, J. M., Some remarks on a paper of S. Cobzas on antiproximinal sets, Bull. Calcutta Math. Soc. 73 (1981), pp. 5–8.Google Scholar
3. Borwein, J. M., Penot, J.-P. and Thera, M., Conjugate convex operators, J. Math. Anal. Appl. 102(1984), pp. 399–414.Google Scholar
4. Bourgin, R., Geometric aspects of convex sets with the Radon-Nikodym property, Lect. Notes in Math. 993, Springer-Verlag, New York, N.Y. (1983).Google Scholar
5. Brøndsted, A. and Rockafellar, R. T., On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), pp. 605–611.Google Scholar
6. Cobzas, S., Antiproximinal sets in some Banach spaces, Math. Balk. 4 (1974), pp. 79–82.Google Scholar
7. Cobzas, S., Convex antiproximinal sets in c and c0 , Mat. Zametki 17 (1975), pp. 449–457.Google Scholar
8. Cobzas, S., Antiproximinal sets in Banach spaces of continuous functions, Anal. Numer. Theor. Approx. 5 (1976), pp. 127–143.Google Scholar
9. Diestel, J. and Uhl, J. J. Jr., Vector measures, Math. Surveys 15, Amer. Math. Soc, Providence, R.I. (1977).Google Scholar
10. Edelstein, M. and Thompson, A. C., Some results on nearest points and support properties of convex sets in c0 , Pacific J. Math. 40 (1972), pp. 553–560.Google Scholar
11. Edelstein, M., Antiproximinal sets, J. Approx. Theory (to appear).Google Scholar
12. Fonf, V. P., Antiproximinal sets in spaces of continuous functions, Math. Notes, Acad. Sci. USSR 33 (1983), pp. 282–287.Google Scholar
13. Ioffe, A. D. and Levin, V. L., Subdifferentials of convex functions, Trans. Moscow Math. Soc. 26(1972), pp. 1–72.Google Scholar
14. Phelps, R. R., Nonexistence of subdifferentials for lower semicontinuous vector-valued convex functions, A.M.S. Abstracts 1 (1980), p. 573, Abstract 80T-B156.Google Scholar
15. Raffin, C., Sur les programmes convexes définis dans des espaces vectoriels topologiques. Annales Inst. Fourier 20 (1970), pp. 457–491.Google Scholar
16. Rubinov, A. M., Sublinear operators and their applications, Russ. Math. Surveys 32 Nr. 4 (1977), pp. 115–175.Google Scholar
17. Valadier, M., Sous-differentiabilité des fonctions convexes à valeurs dans un espace vectoriel ordonné, Math. Scand. 30 (1972), pp. 65–74.Google Scholar
18. Vohra, R., Department of Economies, Brown University, (private communication).Google Scholar
19. Zowe, J., Subdifferentiability of convex functions with values in an ordered vector space, Math. Scand. 34(1974), pp. 69–83.Google Scholar