Let $f$ be a closed concave function on $N_{\mathbb{R}}$. For a subset $C\subset N_{\mathbb{R}}$, the following conditions are equivalent:

1. (1)

   $C\in\Pi(f)$;

2. (2)

   $C=\{u\in N_{\mathbb{R}}\mid x\in\partial f(u)\}$ for a $x\in M_{\mathbb{R}}$;

3. (3)

   there exist $x_{C}\in M_{\mathbb{R}}$ and $\lambda_{C}\in\mathbb{R}$ such that the set $\{(u,\langle x_{C},u\rangle-\lambda_{C})\mid u\in C\}$ is an exposed face of the hypograph of $f$.

In particular, the correspondence

$$C_{x}\mapsto\{(u,\langle x,u\rangle-f^{\vee}(x))\mid u\in C_{x}\}$$

is a bijection between $\Pi(f)$ and the set of non-vertical exposed faces of $\operatorname{hypo}(f)$.