Note: Reposting from OR Stackexchange as advised there.
Consider a convex polyhedron $A$. Assume we have subsets $A_1,\ldots,A_n$ of $A$ that are themselves covex polyhedra and are mutually disjoint except maybe sharing an edge, and that their union gives $A$. A simple example would be $A=[0,10]$ with $A_1=[0,3]$, $A_2=[3,6]$ and $A_3=[6,10]$ (observe $A_1$ and $A_2$ are disjoint except for a common extreme point, and so are $A_2$ and $A_3$, and their union gives $A$).
What do we call this sort of partition-like subdivision of $A$ (in addition to a "cover")?