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We give a short proof of the log-concavity of the coefficients of the reducedcharacteristic polynomial of a matroid. The proof uses an extension of thetheory of Lorentzian polynomials to convex cones, and reproves theHodge-Riemann relations of degree one for the Chow ring of a matroid.

Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture