On the notion of convex-compactness and its applications

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. In particular, we show that bounded-in-probability, convex and closed subsets of the space L+(Ω,F , P) of finite-valued non-negative random variables on a probability space are convex-compact. Applications in optimization and mathematical economics versions of the Minimax theorem, the fixed-point theorem of Knaster, Kuratowski and Mazurkiewicz as well as the excessdemand theorem of mathematical economics are provided.