Let $\Pi$ be a non-empty collection of convex subsets of $N_{\mathbb{R}}$. The collection $\Pi$ is called a convex subdivision if it satisfies the conditions:

1. (1)

   every face of an element of $\Pi$ is also in $\Pi$;

2. (2)

   every two elements of $\Pi$ are either disjoint or they intersect in a common face.

If $\Pi$ satisfies only (2), then it is called a convex decomposition. The support of $\Pi$ is defined as the set $|\Pi|=\bigcup_{C\in\Pi}C$. We say that $\Pi$ is complete if its support is the whole of $N_{\mathbb{R}}$. For a given set $E\subset N_{\mathbb{R}}$, we say that $\Pi$ is a convex subdivision (or decomposition) in $E$ whenever $|\Pi|\subset E$. A convex subdivision in $E$ is called complete if $|\Pi|=E$.