  Hilbert polynomials have positivity properties under favorable conditions. We
establish a similar "K-theoretic positivity" for matroids. As an application,
for a multiplicity-free subvariety of a product of projective spaces such that
the projection onto one of the factors has birational image, we show that a
transformation of its K-polynomial is Lorentzian. This partially answers a
conjecture of Castillo, Cid-Ruiz, Mohammadi, and Montano. As another
application, we show that the h*-vector of a simplicially positive divisor on a
matroid is a Macaulay vector, affirmatively answering a question of Speyer for
a new infinite family of matroids.
