Proposition 7

Let F be a closed subset of X. The following properties are equivalent:

a) F contains $\check{S}_{\mathscr{H}}(X)$.

b) For every function $h \in \mathscr{H}$, the set F intersects the set of points of X where h attains its supremum.

c) For every point $x \in X$, there exists a positive measure $\mu$ of total mass 1 on X, such that $\operatorname{Supp}(\mu) \subset F$ and $h(x) = \int h \, d\mu$ for every function $h \in \mathscr{H}$.

Let $G = i_{\mathscr{H}}(F)$. The condition a) signifies that G contains the set of extremal points of C. The condition b) signifies that G meets the intersection of $i_{\mathscr{H}}(X)$ with each of the closed support hyperplanes of $i_{\mathscr{H}}(X)$. Finally, the condition c) signifies that every point of $i_{\mathscr{H}}(X)$ is the barycenter of a measure with support contained in G; by No. 1, Prop. 1, this is also equivalent to saying that the closed convex envelope of $i_{\mathscr{H}}(X)$ is equal to the closed convex envelope of G. The equivalence of the conditions a), b) and c) therefore follows from TVS, II, §7, No. 1, Cor. of Prop. 2.