The criteria that characterize many interesting classes of lower previsions,
such as coherent or k-monotone lower probabilities, can in finite spaces often
be seen as a set of linear constraints on the set of lower previsions in the
class, which therefore is a convex polyhedron. It can be equivalently
characterized by its set of vertices. For all interesting classes that I
studied, the set of vertices or necessary and sufficient constraints is finite.
In the presentation I aim to make these representations a bit more concrete to
people, so that their possible uses -- both in applications and theory -- can be
discussed in a tangible way.