A weighted inequality for matroid flats and log-concavity of Whitney numbers

Using the Hodge-Riemann relations for matroid Chow rings, we establish a weighted inequality for flats of a loopless matroid. As applications, we prove the points-lines-planes conjecture and Brylawski’s conjecture on the ultra-log-concavity of Whitney numbers of the first kind. We also characterize the cases of equality in both inequalities.