The concave piecewise affine functions and their uniform limits satisfy the following properties.

1. (1)

   Let $f\in{\overline{\mathscr{P}}}_{N_{\mathbb{R}}}$. Then $f^{\vee\vee}=f$.

2. (2)

   If $f\in\mathscr{P}(\Lambda,\Lambda^{\prime})$ (respectively $f\in{\overline{\mathscr{P}}}(\Lambda,\Lambda^{\prime})$) then $f^{\vee}\in\mathscr{P}(\Lambda^{\prime},\Lambda)$ (respectively $f^{\vee}\in{\overline{\mathscr{P}}}(\Lambda^{\prime},\Lambda)$).

3. (3)

   If $f\in{\overline{\mathscr{P}}}(\Lambda)$ then ${\operatorname{dom}}(\operatorname{rec}(f))=\operatorname{rec}(\Lambda)$.

4. (4)

   Let $f_{i}\in\mathscr{P}(\Lambda_{i},\Lambda_{i}^{\prime})$ (respectively $f_{i}\in{\overline{\mathscr{P}}}(\Lambda_{i},\Lambda_{i}^{\prime})$), $i=1,2$, with $\Lambda_{1}\cap\Lambda_{2}\not=\emptyset$. Then $f_{1}+f_{2}\in\mathscr{P}(\Lambda_{1}\cap\Lambda_{2},\Lambda_{1}^{\prime}+\Lambda_{2}^{\prime})$ (respectively $f_{1}+f_{2}\in{\overline{\mathscr{P}}}(\Lambda_{1}\cap\Lambda_{2},\Lambda_{1}^{\prime}+\Lambda_{2}^{\prime})$) and $(f_{1}+f_{2})^{\vee}=f_{1}^{\vee}\boxplus f_{2}^{\vee}$.

5. (5)

   Let $f_{i}\in\mathscr{P}(\Lambda_{i},\Lambda_{i}^{\prime})$ (respectively $f_{i}\in{\overline{\mathscr{P}}}(\Lambda_{i},\Lambda_{i}^{\prime})$), $i=1,2$, with $\Lambda_{1}^{\prime}\cap\Lambda_{2}^{\prime}\not=\emptyset$. Then $f_{1}\boxplus f_{2}\in\mathscr{P}(\Lambda_{1}+\Lambda_{2},\Lambda_{1}^{\prime}\cap\Lambda_{2}^{\prime})$ (respectively $f_{1}+f_{2}\in{\overline{\mathscr{P}}}(\Lambda_{1}+\Lambda_{2},\Lambda_{1}^{\prime}\cap\Lambda_{2}^{\prime})$) and $(f_{1}\boxplus f_{2})^{\vee}=f_{1}^{\vee}+f_{2}^{\vee}$.

6. (6)

   Let $(f_{i})_{i\geq 1}\subset{\overline{\mathscr{P}}}$ be a sequence converging uniformly to a function $f$. Then $f\in{\overline{\mathscr{P}}}$.