Corollary

If the closed convex envelope C of K in E is compact, then the barycenter of every positive measure of total mass 1 on K belongs to E.

For, C is then also the closed convex envelope of K in ${E'}^*$ equipped with the weak topology $\sigma(E', E')$, and it suffices to apply, to the canonical injection of K into E, Prop. 4 of Ch. III, §3, No. 2.