Ehrhart theory studies the polynomial function that counts lattice points in integer dilates of a lattice polytope. This talk will focus on a natural mixed extension: counting lattice points in Minkowski sums of several scaled lattice polytopes. This produces a multivariate polynomial whose coefficients in the binomial basis are known as discrete mixed volumes, which generalize the classical mixed volumes in the case of lattice polytopes. Mixed volumes famously satisfy the Alexandrov–Fenchel inequalities, which has been used to prove various log-concavity results in combinatorics. A natural question is whether analogous inequalities hold for the discrete mixed volumes. I will present an asymptotic Alexandrov–Fenchel–type inequality valid for general families of lattice polytopes, and an exact inequality in the case of coordinate simplices, where the proof relies on an exceptional version of the Hirzebruch–Riemann–Roch theorem on the permutohedral variety due to Berget–Eur–Spink–Tseng. We will also see that such inequalities fail in general.