  We investigate a recent semantics for intermediate (and modal) logics in
terms of polyhedra. The main result is a finite axiomatisation of the
intermediate logic of the class of all polytopes -- i.e., compact convex
polyhedra -- denoted PL. This logic is defined in terms of the Jankov-Fine
formulas of two simple frames. Soundness of this axiomatisation requires
extracting the geometric constraints imposed on polyhedra by the two formulas,
and then using substantial classical results from polyhedral geometry to show
that convex polyhedra satisfy those constraints. To establish completeness of
the axiomatisation, we first define the notion of the geometric realisation of
a frame into a polyhedron. We then show that any PL frame is a p-morphic image
of one which has a special form: it is a 'sawed tree'. Any sawed tree has a
geometric realisation into a convex polyhedron, which completes the proof.
