Whitney numbers of arrangements via measure concentration of intrinsic volumes

TL;DR

This paper proves the Rota-Heron-Welsh conjecture for a broad class of matroids by linking intrinsic volumes with measure concentration phenomena, advancing understanding of matroid characteristic polynomials.

Contribution

It introduces a novel approach combining intrinsic volumes and measure concentration to verify log-concavity of characteristic polynomial coefficients for realizable c-arrangements.

Findings

Confirmed the Rota-Heron-Welsh conjecture for c-arrangements

Extended the class of matroids for which log-concavity is proven

Connected geometric measure theory with combinatorial properties of matroids

Abstract

We verify the Rota-Heron-Welsh conjecture for matroids realizable as c-arrangements: the coefficients of the characteristic polynomial of the associated matroid are log-concave. This family of matroids strictly contains that of complex hyperplane arrangements. Our proof combines the study of intrinsic volumes of certain extensions of arrangements and the Levy--Milman measure concentration phenomenon on realization spaces of arrangements.