Exercise 11, § 7

a) In the space $\mathbf{R}^3$, consider the compact convex set A which is the convex envelope of the union of the circle $\zeta = 0, \xi^2 + \eta^2 - 2\xi = 0$ and the two points $(0, 0, 1)$ and $(0, 0, -1)$. Show that the set of extremal points of A is not closed in A. b) Let A be a metrisable compact convex set in a Hausdorff topological vector space E. Show that the set of extremal points of A is the intersection of a sequence of open sets in A. (If d is a distance defining the topology of A, then for each integer n consider the set of points $x = \frac{1}{2}(y + z)$, where y, z are in A and $d(y, z) \geq 1/n$.)