Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T12:26:56.390Z Has data issue: false hasContentIssue false
  • English
  • Français

SEPARATION OF CONVEX SETS IN EXTENDED NORMED SPACES

Part of: Topological linear spaces and related structures Normed linear spaces and Banach spaces; Banach lattices General convexity

Published online by Cambridge University Press:  26 February 2015

G. BEER  and
J. VANDERWERFF
G. BEER
Affiliation:
Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, CA 90032, USA email gbeer@cslanet.calstatela.edu
J. VANDERWERFF*
Affiliation:
Department of Mathematics, La Sierra University, 4500 Riverwalk Parkway, Riverside, CA 92515, USA email jvanderw@lasierra.edu
*
✉jvanderw@lasierra.edu
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.

We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$, that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

Keywords

extended normed spaceconvex setcontinuous separationproper separationstrict separationstrong separationcoreconvex cone

MSC classification

Primary: 46A55: Convex sets in topological linear spaces; Choquet theory
Secondary: 46B20: Geometry and structure of normed linear spaces 52A05: Convex sets without dimension restrictions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 2 , October 2015 , pp. 145 - 165
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Beer, G., ‘The structure of extended real-valued metric spaces’, Set-Valued Var. Anal. 21 (2013), 591–602.CrossRefGoogle Scholar
Beer, G., Costantini, C. and Levi, S., ‘Bornological convergence and shields’, Mediterr. J. Math. 10 (2013), 529–560.CrossRefGoogle Scholar
Beer, G., ‘Norms with infinite values’, J. Convex Anal. 22 (2015), 35–58.Google Scholar
Beer, G. and Hoffman, M., ‘The Lipschitz metric for real-valued continuous functions’, J. Math. Anal. Appl. 406 (2013), 229–236.CrossRefGoogle Scholar
Beer, G. and Levi, S., ‘Uniform continuity, uniform convergence, and shields’, Set-Valued Var. Anal. 18 (2010), 251–275.CrossRefGoogle Scholar
Borwein, J. and Vanderwerff, J., Convex Functions: Constructions, Characterizations and Counterexamples, Encyclopedia of Mathematics and its Applications, 109 (Cambridge University Press, Cambridge, 2010).CrossRefGoogle Scholar
Fabian, M., Habala, P., Hájek, P., Montesinos, V., Pelant, J. and Zizler, V., Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics, 8 (Springer, New York, 2001).CrossRefGoogle Scholar
Holmes, R., Geometric Functional Analysis and its Applications (Springer, New York, 1975).CrossRefGoogle Scholar
Rockafellar, R. T., Convex Analysis (Princeton University Press, Princeton, NJ, 1970).CrossRefGoogle Scholar
Valentine, F. A., Convex Sets (McGraw-Hill, New York, 1964).Google Scholar
Zalinescŭ, C., Convex Analysis in General Vector Spaces (World Scientific, River Edge, NJ, 2002).CrossRefGoogle Scholar