Connectivity Functions and Polymatroids

A {\em connectivity function on} a set \(E\) is a function \(\lambda:2^E\rightarrow \mathbb R\) such that \(\lambda(\emptyset)=0\), that \(\lambda(X)=\lambda(E-X)\) for all \(X\subseteq E\) and that \(\lambda(X\cap Y)+\lambda(X\cup Y)\leq \lambda(X)+\lambda(Y)\) for all \(X,Y \subseteq E\). Graphs, matroids and, more generally, polymatroids have associated connectivity functions. We introduce a notion of duality for polymatroids and prove that every connectivity function is the connectivity function of a self-dual polymatroid. We also prove that every integral connectivity function is the connectivity function of a half-integral self-dual polymatroid.