Matroids, log-concavity and measure concentration

We provide a simple proof of the Rota--Heron--Welsh conjecture for matroids realizable as c-arrangements in the sense of Goresky--MacPherson: we prove that the coefficients of the characteristic polynomial of the associated matroids form log-concave sequences, proving the conjecture for a family of matroids out of reach for all previous methods.

To this end, we study the L\'evy--Milman measure concentration phenomenon on natural push-forwards of uniform measures on the Grassmannian to realization spaces of arrangements under a certain extension procedure on matroids.