  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.
