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Mathematics and Financial Economics

, Volume 3, Issue 1, pp 1–12 | Cite as

Convex compactness and its applications

  • Gordan Žitković
Article

Abstract

The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in lieu of compactness in a variety of cases. Specifically, we establish convex compactness for certain familiar classes of subsets of the set of positive random variables under the topology induced by convergence in probability. Two applications in infinite-dimensional optimization—attainment of infima and a version of the Minimax theorem—are given. Moreover, a new fixed-point theorem of the Knaster-Kuratowski-Mazurkiewicz-type is derived and used to prove a general version of the Walrasian excess-demand theorem.

Keywords

Convex compactness Excess-demand theorem Fréchet spaces KKM theorem Minimax theorem Optimization 

JEL Classification

C61 C62 

Mathematics Subject Classification (2000)

Primary 46N10 Secondary 47H10 60G99 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA

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