Exercise 23, § 7

In $\mathbf{R}^3$, consider a closed convex set A with an interior point, whose fonction F contains two open segments S, T lying in two non-parallel lines D, D' (the points of S, T are thus non-extremal in A), and all of whose other frontier points are extremal (one shows how to define such convex sets). For every $x \in \mathbf{R}^3$, put $f(x) = (d(x, D))^2$. Let B the convex closed set in $\mathbf{R}^4 = \mathbf{R}^3 \times \mathbf{R}$ formed by the pairs $(x, \zeta)$ such that $x \in A$ and $\zeta \geq f(x)$. Show that in B the set of extremal points, the union of the extremal rays, the set of end points of extremal rays, and all the unions of two or three of these sets are not closed and non-empty.