ON EMBEDDING A COMPACT CONVEX SET INTO A LOCALLY CONVEX TOPOLOGICAL VECTOR SPACE

If a compact convex subset of a topological vector space is strongly locally convex then it is affinely homeomorphic to a convex subset of a locally convex topological vector space. Throughout this paper, a convex set X will be a convex subset of a real Hausdorff topological vector space endowed with the relative topology. If the relative topology is (locally) compact we will say that X is a (locally) compact convex set. Usually, when one proves results about compact convex sets, for example in the Krein-Milman and Ryll-Nardzewski Theorems, one requires that the containing vector space be locally convex. It is not known whether this requirement is necessary. It would not be neces-sary, for example, if every compact convex set could be embedded by an affine homeomorphism into a locally convex topological vector space. But it is not known whether such embeddings always exist.1 In this paper we show that if a compact convex set satisfies a certain loca