Whitney numbers via measure concentration in representation varieties

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 Lévy--Milman measure concentration phenomenon on natural pushforwards of uniform measures on the Grassmannian to realization spaces of arrangements under a certain extension procedure on matroids.