On convex (concave) roofs

Given a function g on the exrtremal points of a convex compact set K, one may ask for the largest convex extension $g^{\cup}$ of $g$ on $K$. Similar the smallest concave extension $g^{\cap}$ ist defined. Entanglement of formation, entanglement of assistance, some concurrences belong to these classes of functions. If the extreme boundary of K is also compact and g is continuous, then a remarkable phenomenon takes place: $K$ is covered by convex subsets on each of which the extension, say $g^{\cup}$, is affine, and every one of these subsets is convexly generated by some extremal points of $K$ . Because of this I have called $g^{\cup}$ and $g^{\cap}$ "roofs" . I try to explain how to use this as a tool and I show the structure of some of the known examples, with and without bifurcations.