Let $\Delta$ be a convex polytope of $M_{\mathbb{R}}$.

1. (1)

   The space ${\overline{\mathscr{P}}}(\Delta,N_{\mathbb{R}})$ agrees with the space of all continuous concave functions on $\Delta$.

2. (2)

   A concave function $f$ belongs to ${\overline{\mathscr{P}}}(N_{\mathbb{R}},\Delta)$ if and only if ${\operatorname{dom}}(f)=N_{\mathbb{R}}$ and $|f-\Psi_{\Delta}|$ is bounded.