  We recover the Tutte polynomial of a matroid, up to change of coordinates,
from an Ehrhart-style polynomial counting lattice points in the Minkowski sum
of its base polytope and scalings of simplices. Our polynomial has coefficients
of alternating sign with a combinatorial interpretation closely tied to the
Dawson partition. Our definition extends in a straightforward way to
polymatroids, and in this setting our polynomial has K\'alm\'an's internal and
external activity polynomials as its univariate specialisations.
