Exercise 9, § 6

Let F and G be two vector spaces in separating duality and A be a convex subset of F which contains 0. For every $y \in G$, write

so that $0 \leq H_A(y) \leq +\infty$; we call $H_A$ the support function of A. a) Show that $H_A$ is the gauge of $A^\circ$ (II, p. 20). b) If A is weakly compact, then, for all $y \in G$, the hyperplane with the equation $\langle x, y \rangle = H_A(y)$ is a support hyperplane of A. c) $H_A$ is finite and continuous for the topology $\sigma(G, F)$ if, and only if, A is finite-dimensional and bounded (in the finite dimensional vector subspace that it generates). d) Let $A_i$ ($1 \leq i \leq p$) be convex sets in F which contain 0 and $\lambda_i$ be real numbers $\geq 0$ ($1 \leq i \leq p$); show that the support function of the convex set $A = \sum_i \lambda_i A_i$ is $H_A = \sum_i \lambda_i H_{A_i}$. If $y \in G$ is such that the intersection $C_i$, of $A_i$ with the hyperplane $\langle x, y \rangle = H_A(y)$, is non-empty for $1 \leq i \leq p$, show that the intersection of A and the hyperplane with equation $\langle x, y \rangle = H_A(y)$ is the set $\sum_i \lambda_i C_i$. e) Suppose that A is locally compact and does not contain any line. Then the set which is the union of $\{0\}$ and the set of $y \neq 0$ such that $H_A(y) = +\infty, H_A(-y) \neq +\infty$ is the polar cone of the asymptotic cone $C_A$ (consider the case when A is the convex envelope of $\{0\}$ and of one half-line). f) Suppose that F is finite dimensional. Show that, A is parabolic (II, p. 67, exerc. 17) if and only if $H_A$ is a continuous mapping of G in $\overline{\mathbf{R}}$ (if there exists a line paralle to a half-line of $C_A$, which does not meet A, note that there exists a hyperplane separating this half-line from A).