The Funk metric is an asymmetric metric related to, but somewhat simpler than, the Hilbert metric. We are interested in the volume of balls in this geometry, particularly in the case where the domain is a polytope.

Our study reveals a new link between the Mahler conjecture and some known conjectures concerning the combinatorics of polytopes. We are also led to define a new affine invariant for polytopes that seems to be an analogue of the centro-affine area.

This is joint work with Dmitry Faifman and Constantin Vernicos.