Exercise 10, § 6

To each compact convex set A in $E = \mathbf{R}^n$ containing 0, we make correspond its support function $H_A$ by the duality between E and $E^* : H_A$ belonging to the space $\mathscr{C}(E^* ; \mathbf{R})$ of continuous real valued functions in $E^*$. We ascribe to the space $\mathscr{C}(E^* ; \mathbf{R})$ the uniform structure of compact convergence and, to the set $\mathfrak{K}_0'(E)$ of the compact convex sets in E containing 0, the uniform structure defined in the exerc. 39 of II, p. 71. Show that $A \mapsto H_A$ is an isomorphism of $\mathfrak{K}_0'(E)$ on a uniform subspace of $\mathscr{C}(E^* ; \mathbf{R})$. Deduce that the mapping $A \mapsto A^\circ$ of the set $\mathfrak{K}_0(E)$ of compact convex sets in E which contain 0 as an interior point, on the set $\mathfrak{K}_0(E^*)$, is an isomorphism for the uniform structures of these two spaces (cf. II, p. 71, exerc. 39).