Convex-compactness and its applications

TL;DR

This paper introduces the concept of convex compactness, a weaker form of compactness, and demonstrates its applications in infinite-dimensional optimization, fixed-point theorems, and economic equilibrium analysis.

Contribution

It defines convex compactness, shows its prevalence in convex subsets of topological vector spaces, and applies it to optimization, fixed points, and economic theorems.

Findings

Convex compactness applies to many convex subsets in topological vector spaces.

Established convex compactness for positive random variables under convergence in probability.

Derived a new fixed-point theorem and proved a general Walrasian excess-demand theorem.

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.