Embeddings of compact convex sets and locally compact cones
Jimmie Lawson · Pacific Journal of Mathematics · 1976
The main result of this paper is that a compact convex set with a basis of neighborhoods (not necessarily open) at each point which are convex can be embedded in a locally convex separated topological vector space.An analogous result is proved for locally compact cones.Along the way it is shown that any compact convex set can be embedded as a base of a locally compact cone in a separated topological vector space, and that the various notions of local convexity coincide in a compact convex set.PROPOSITION 2.1.Λ compact topological convex set is affinely homeomorphic to a base of a locally compact cone.Proof.Let X be a compact convex set in some real vector space V. Define a new vector space W = R x V, and setIt is straightforward to show using the convexity of X that C is a cone