Matroids and log-concavity

We show that f -vectors of matroid complexes of realizable matroids are strictly log-concave. This was conjectured by Mason in 1972. Our proof uses the recent result by Huh and Katz who showed that the coefficients of the characteristic polynomial of a realizable matroid form a log-concave sequence. We also prove a statement on log-concavity of h-vectors which strengthens a result by Brown and Colbourn. In the last two sections, we give a brief introduction to zonotopal algebra and we explain how it relates to our log-concavity results and various matroid/graph polynomials.