Exercise 34, § 7

Let S be a closed convex set of E, a Hausdorff locally convex space, and A a subset of S such that S = Γ_0(A) (exerc. 33), and let S_0 be the convex envelope of A (so that S = S̄_0). Let N be a closed convex subset of E containing a closed linear variety of finite codimension, M = S ∩ N and M_0 = S_0 ∩ N. a) Show that M = M̄_0. (Note, using exerc. 33, a), that every closed linear variety of finite codimension in E, containing a point x ∈ M, meet M_0, and use the prop. 4 of II, p. 38.) b) Suppose that for every finite subset F of A, the intersection of N and of the facet (in S) of each point of the convex envelope of F, is compact or of finite dimension and does not contain a line. Show then that M is the convex closed envelope of the set of its extremal points. (By the aid of a), this reduces to proving that every point of M_0 is contained in the convex closed envelope of the set of extremal points of M. Use exerc. 3, e) of II, p. 87, the Krein-Milman th. (II, p. 55) and exerc. 22, b) of II, p. 90.) Deduce that every closed support hyperplane of M contains an extremal point of M.