Proposition 4

(i) The closed convex envelope C of $i_{\mathscr{H}}(X)$ in $\mathscr{H}'$ (for the weak topology $\sigma(\mathscr{H}', \mathscr{H})$) is compact.

(ii) For a point $i_{\mathscr{H}}(x)$ to be an extremal point of C, it is necessary and sufficient that the only positive measure $\lambda$ on X such that

for every function $h \in \mathscr{H}$ (which implies in particular that $\lambda$ has total mass 1, since $1 \in \mathscr{H}$) be the Dirac measure $\varepsilon_x$.

It follows that, for each $h \in \mathscr{H}$, the function $z' \mapsto \langle h, z' \rangle$ on C attains its supremum at at least one extremal point of C (TVS, II, §7, No. 1, Prop. 1), and this point belongs to $i_{\mathscr{H}}(X)$ (loc. cit., Cor. of Prop. 2).

(i) By (9), $\| i_{\mathscr{H}}(x) \| \leq 1$ in the normed space $\mathscr{H}'$, in other words $i_{\mathscr{H}}(X)$ is bounded, and the assertion follows from the fact that $\mathscr{H}'$, equipped with the weak topology $\sigma(\mathscr{H}', \mathscr{H})$, is quasi-complete (TVS, III, §4, No. 2, Cor. 5 of Th. 1).

(ii) Every positive measure $\mu$ of mass 1 on $i_{\mathscr{H}}(X)$ arises, by transport of structure by means of the homeomorphism $i_{\mathscr{H}}$, from a positive measure $\lambda$ of mass 1 on X, the Dirac measure $\varepsilon_{i_{\mathscr{H}}(x)}$ arising from $\varepsilon_x$. To say that $\mu$ admits $i_{\mathscr{H}}(x)$ as barycenter means, by definition, that

for every function $h \in \mathscr{H}$. Taking (9) into account, the assertion (ii) is just the translation of the criterion of No. 2, Cor. of Prop. 3 for $i_{\mathscr{H}}(x)$ to be an extremal point of C.

We shall say that a point $x \in X$ satisfying condition (ii) of Prop. 4 is $\mathscr{H}$-extremal; we denote by $\mathrm{Ch}_{\mathscr{H}}(X)$ (or simply $\mathrm{Ch}(X)$) the set of these points, and by $\check{S}_{\mathscr{H}}(X)$ (or simply $\check{S}(X)$) the closure of $\mathrm{Ch}_{\mathscr{H}}(X)$ in $X$.