Exercise 22, § 7

By an extremal ray of a convex subset A of a vector space E we mean a closed half line D contained in A, such that, for all $x \in D$ and every open segment with end points $a, b$ in A, and which contains x, it is necessarily the case that $a \in D$ and $b \in D$; the end point of D is an extremal point of A. a) In a Hausdorff locally convex space E, show that every locally compact closed convex set, not containing a line, is the closed convex envelope of the union of its extremal rays and its extremal points. (Suppose the contrary, and writing B for this closed convex envelope, note first that by exerc. 21, d), there exists a closed hyperplane H so that $H \cap A$ is compact and non-empty and $H \cap B = \varnothing$. Show then that if $a \in H \cap A$ is an extremal point of $H \cap A$ (therefore not an extremal point of A by hypothesis) and if the open segment S with end points $b, c$ contained in A and not contained in H, contains $a$, then the line D containing S necessarily contains a segment containing $a$ and whose end points are extremal points of A, or contains an extremal ray of A containing $a$. b) Prove that if E is finite dimensional, then every closed convex set in E, that does not contain any line, is the convex envelope of the union of its extremal points and its extremal rays (argue by induction on the dimension of E).