  We discuss aspects of the convex-roof extension of multipartite entanglement
measures, that is, $SL(2,\CC)$ invariant tangles. We highlight two key concepts
that contain valuable information about the tangle of a density matrix: the
{\em zero-polytope} is a convex set of density matrices with vanishing tangle
whereas the {\em convex characteristic curve} readily provides a non-trivial
lower bound for the convex roof and serves as a tool for constructing the
convex roof outside the zero-polytope. Both concepts are derived from the
tangle for superpositions of the eigenstates of the density matrix. We
illustrate their application by considering examples of density matrices for
two-qubit and three-qubit states of rank 2, thereby pointing out both the power
and the limitations of the concepts.
