Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-06T18:40:57.677Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-concave functions and convex convergence to infinity

Published online by Cambridge University Press:  17 April 2009

Gerald Beer
Gerald Beer
Affiliation:
Department of MathematicsCalifornia State UniversityLos Angeles CA 90032United States of America
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.

By a convex mode of convergence to infinity 〈Ck〉, we mean a sequence of nonempty closed convex subsets of a normed linear space X such that for each k, Ck+1 ⊆ int Ck and and a sequence 〈xn〉 is X is declared convergent to infinity with respect to 〈Ck〉 provided each Ck contains xn eventually. Positive convergence to infinity with respect to a pointed cone with nonempty interior as well as convergence to infinity in a fixed direction fit within this framework. In this paper we study the representation of convex modes of convergence to infinity by quasi-concave functions and associated remetrizations of the space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 1 , August 1999 , pp. 81 - 94
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Beer, G., Topologies on closed and closed convex sets (Kluwer Academic Publishers, Dordrecht, Holland, 1993).CrossRefGoogle Scholar
[2]Beer, G., ‘On metric boundedness structures’, Set-Valued Anal, (to appear).Google Scholar
[3]Borwein, J., Penot, J.-P. and Thera, M., ‘Conjugate convex operators’, J. Math. Anal. Appl. 102 (1984), 399–414.CrossRefGoogle Scholar
[4]Deimling, K., Nonlinear functional analysis (Springer-Verlag, Berlin, Heidelberg, New York, 1985).CrossRefGoogle Scholar
[5]Dugundji, J., Topology (Allyn and Bacon, Boston, 1966).Google Scholar
[6]Franchetti, C. and Singer, I., ‘Best approximation by elements of caverns in normed linear spaces’, Boll. Un. Mat. Ital. (5) 17 (1980), 33–43.Google Scholar
[7]Hadwiger, H.. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer-Verlag, Berlin, Heidelberg, New York, 1957).CrossRefGoogle Scholar
[8]Hiriart-Urruty, J.-B., ‘Tangent cones, generalized gradients, and mathematical programming in Banach spaces’, Math. Oper. Res. 4 (1979), 79–97.CrossRefGoogle Scholar
[9]Hiriart-Urruty, J.-B., ‘New concepts in nondifferentiable programming’, Bull. Soc. Math. France Mem. 60 (1979), 57–85.Google Scholar
[10]Hiriart-Urruty, J.-B. and Lemaréchal, C., Convex analysis and minimization algorithms I (Springer-Verlag, Berlin, Heidelberg, New York, 1993).CrossRefGoogle Scholar
[11]Holmes, R., A course in optimization and best approximation, Lecture Notes in Mathematics 257 (Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[12]Hu, S.-T., Introduction to general topology (Holden-Day, San Francisco, 1966).Google Scholar
[13]Lemaire, B., ‘Duality in reverse convex optimization’, SIAM J. Optimization (to appear).Google Scholar
[14]Sangwine-Yager, J., ‘Bonnesen-style inequalities for Minkowski relative geometry’, Trans. Amer. Math. Soc. 307 (1988), 373–382.CrossRefGoogle Scholar
[15]Sion, M., ‘On general minimax theorems’, Pacific J. Math. 8 (1958), 171–176.CrossRefGoogle Scholar
[16]Van Tiel, J.. Convex analysis (J. Wiley and Sons. New York, 1984).Google Scholar