We start by proving (1). By the properties of uniform convergence, it is clear that any element of ${\overline{\mathscr{P}}}(\Delta,N_{\mathbb{R}})$ is concave and continuous. Conversely, a continuous function $f$ on $\Delta$ is uniformly continuous because $\Delta$ is compact. Therefore, given $\varepsilon>0$ there is a $\delta>0$ such that $|f(u)-f(v)|<\varepsilon$ for all $u,v\in\Delta$ such that $\|u-v\|<\delta$. By compactness, we can find a triangulation $\Delta=\bigcup_{i}\Delta_{i}$ with $\operatorname{diam}(\Delta_{i})<\delta$. Let $\{b_{j}\}_{j}$ be the vertices of this triangulation and consider the function $g\in\mathscr{P}(\Delta,N_{\mathbb{R}})$ defined as

$$g(u)=\sup\bigg\{\sum_{j=1}^{l}\lambda_{j}f(b_{j})\bigg|\ \lambda_{j}\geq 0,\sum_{j}\lambda_{j}=1\sum_{j}\lambda_{j}a_{j}=x\bigg\}.$$

For $u\in\Delta$, let $b_{j_{0}},\dots,b_{j_{n}}$ denote the vertices of an element of the triangulation containing $u$. We write $u=\lambda_{j_{0}}u_{j_{0}}+\dots+\lambda_{j_{n}}u_{j_{n}}$ for some $\lambda_{j_{i}}\geq 0$ and $\lambda_{j_{0}}+\dots+\lambda_{j_{n}}=1$. By concavity, we have

$$f(u)\geq g(u)\geq\sum_{k=0}^{n}\lambda_{j_{k}}f(u_{j_{k}})\geq f(u)-\varepsilon,$$

which shows that any continuous function on $\Delta$ can be arbitrarily approximated by elements of $\mathscr{P}(\Delta,N_{\mathbb{R}})$.

We now prove (2). Let $f\in{\overline{\mathscr{P}}}(N_{\mathbb{R}},\Delta)$. By definition, for each $\varepsilon>0$ we can find a function $g\in\mathscr{P}(N_{\mathbb{R}},\Delta)$ with $\sup|f-g|\leq\varepsilon$. In particular, $|f-g|$ is bounded. Furthermore, $\operatorname{rec}(g)=\Psi_{\Delta}$ and $|g-\operatorname{rec}(g)|$ is bounded because $g\in\mathscr{P}(N_{\mathbb{R}})$. Hence ${\operatorname{dom}}(f)={\operatorname{dom}}(g)=N_{\mathbb{R}}$ and $|f-\Psi_{\Delta}|$ is bounded.

Conversely, let $f$ be a concave function such that ${\operatorname{dom}}(f)=N_{\mathbb{R}}$ and $|f-\Psi_{\Delta}|$ is bounded. Then $\operatorname{stab}(f)=\operatorname{stab}(\Psi_{\Delta})=\Delta$ and $f^{\vee}$ is a continuous concave function on $\Delta$. Hence we can apply (1) to $f^{\vee}$ to obtain functions $g_{i}\in\mathscr{P}(\Delta,N_{\mathbb{R}})$ approaching $f^{\vee}$ uniformly. We conclude that the functions $g_{i}^{\vee}\in\mathscr{P}(N_{\mathbb{R}},\Delta)$ approach $f$ uniformly and so $f\in{\overline{\mathscr{P}}}(N_{\mathbb{R}},\Delta)$. ∎