A Characterization of Polyhedral Convex Sets

This paper describes a class of convex closed sets, S, in Rn for which the following property holds: for every correspondence defined on a probability space with relative open values in S its integral is a relative open subset of S. It turns out, that the only closed convex sets in Rn having this property are generalized polyhedral convex sets. In particular, the only compact convex sets in Rn having this property are polytopes.