Exercise 18, § 6

a) In a normed space E, let A be a closed convex set $\neq E$; show that the function $x \mapsto d(x, \complement A)$ is concave in A (use the fact that A is the intersection of closed half-spaces). b) Define inductively a sequence of closed convex sets $A_n \subset \mathbf{R}^n$ in the following manner; $A_1 = \mathbf{R}_+$; if $\mathbf{R}^{n+1}$ is identified with $\mathbf{R}^n \times \mathbf{R}$, then $A_{n+1}$ is the set of pairs $(x, \xi)$ such that $x \in \overset{\circ}{A}_n$ and that

where $\|x\|$ is the Euclidean norm. Show that $A_{n+1}$ does not have any support hyperplane of the form $H \times \mathbf{R}$, where H is a hyperplane of $\mathbf{R}^n$ and that its asymptotic cone is $\{0\} \times \mathbf{R}_+$. c) If $p_{nm}$ is the canonical projection $\mathbf{R}^m \to \mathbf{R}^n$ ($\mathbf{R}^m$ being identified with $\mathbf{R}^n \times \mathbf{R}^{m-n}$) for $m \geq n$, show that when $\mathbf{R}^N$ is identified with the projective limit of the projective system $(\mathbf{R}^n, p_{nm})$ the $A_n$ form a projective system of sets and that $A = \lim \leftarrow A_n$ is a closed convex set not relatively compact in $\mathbf{R}^N$, having no closed hyperplane of support and such that $C_A = \{0\}$.