  For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the
mixed intersection numbers of certain classes in the combinatorial Chow ring
$A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann
relations, we deduce a strengthening of the log-concavity of the $h$-vector of
a matroid complex, improving on an old conjecture of Dawson.
