In a recent joint work with June Huh, we proved the log concavity of the characteristic polynomial of a realizable matroid by relating its coefficients to intersection numbers on an algebraic variety and applying an algebraic geometric inequality. This extended earlier work of Huh that resolved a conjecture in graph theory. In this talk, we rephrase the problem in terms of intersection theory on particular polyhedral complexes, called tropical surfaces. We outline a possible approach to the general log concavity question for all matroids in terms of a conjectural analogue of the Hodge index theorem. This leads us to new polygonal invariants of matroids called `combinatorial Okounkov bodies’ and an equivalent conjecture regarding their areas.