  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.
