Let $E,E^{\prime}$ be subsets of $N_{\mathbb{R}}$ and $M_{\mathbb{R}}$ respectively, and $\Pi,\Pi^{\prime}$ convex decompositions of $E$ and $E^{\prime}$, respectively. We say that $\Pi$ and $\Pi^{\prime}$ are dual convex decompositions if there exists a bijective map $\Pi\to\Pi^{\prime},C\mapsto C^{\ast}$ such that

1. (1)

   for all $C,D\in\Pi$ we have $C\subset D$ if and only if $C^{\ast}\supset D^{\ast}$;

2. (2)

   for all $C\in\Pi$ the sets $C$ and $C^{\ast}$ are contained in orthogonal affine spaces of $N_{\mathbb{R}}$ and $M_{\mathbb{R}}$, respectively.