Consider a correspondence $C: X \rightrightarrows Y$, that is defined as $C(x) \doteq \{ f_s(x) \mid s \in S \}$ where for all $s \in S$, $f_s$ is a concave function.
Can I re-express this correspondence in the following form $C^\prime(x) \doteq \{y \in Y | g_1(x,y), …, g_n(x,y) \geq 0 \}$ for some functions, $g_i: X \times Y \to R$ for all $\forall i \in [n]$.