A Characterization of Polyhedral Convex Sets

This paper describes a class of convex closed sets, S, in

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

R^n R n

having this property are generalized polyhedral convex sets. In particular, the only compact convex sets in

R^n R n

having this property are polytopes.