# Base polyhedron

Let V be a finite ground set, and let $b:2^V \to {\mathbb R} \cup \{\infty\}$ be a submodular function for which b(V) is finite and $b(\emptyset) = 0$. The base polyhedron of b is
$B(b)=\{x \in {\mathbb R}^V: x(Z) \leq b(Z)\ \forall Z\subseteq V,\ x(V)=b(V)\}.$
The same definition can be used for a crossing submodular function b. In this case, the polyhedron may be empty, and non-emptiness is characterized by Fujishige's base polyhedron theorem. However if it is non-empty, then there is a submodular function b' for which $B(b)=B(b')$.