Exercise 36, § 2

Let A be a compact convex set in $\mathbf{R}^n$ and B its projection on the subspace $\mathbf{R}^{n-1}$ (identified as the hyperplane with equation $\xi_n = 0$). Show that there exist two convex functions $f_1, f_2$ defined over B, such that A is identical with the set of points $(x, \zeta)$ of $\mathbf{R}^n$ where $x \in B, y \in \mathbf{R}$ and $f_1(x) \leq \zeta \leq -f_2(x)$.