Exercise 37, § 2

Let E be a vector space; in order that a convex set F of $E \times \mathbf{R}$ should be formed of pairs $(x, \zeta)$ such that $f(x) \leq \zeta$ (resp. $f(x) < \zeta$) for a convex function $f$ defined over a convex subset X of E, it is necessary and sufficient that the projection of F on E should be identical with X and that, for all $x \in X$, the set $F(x)$ of F that projects onto x should be a closed (resp. open) interval unlimited to the right ($i.e.$ not bounded above).