Let $C,C^{\prime}$ be convex polyhedra in $N_{\mathbb{R}}$ and $M_{\mathbb{R}}$, respectively, and $\Pi,\Pi^{\prime}$ polyhedral complexes in $C$ and $C^{\prime}$, respectively. We say that $\Pi$ and $\Pi^{\prime}$ are dual polyhedral complexes if there is a bijective map $\Pi\to\Pi^{\prime},\Lambda\mapsto\Lambda^{*}$ such that

1. (1)

   for all $\Lambda,K\in\Pi$, the inclusion $K\subset\Lambda$ hols if and only if $K^{*}\supset\Lambda^{*}$;

2. (2)

   for all $\Lambda,K\in\Pi$, if $K\subset\Lambda$, then $\angle(\Lambda^{*},K^{*})=\angle(K,\Lambda)^{\vee}$.