Spanning Lattice Polytopes and the Uniform Position Principle

A lattice polytope $P$ is called IDP if any lattice point in its $k$th dilate is a sum of $k$ lattice points in $P$. In 1991 Stanley proved a strong inequality in Ehrhart theory for IDP lattice polytopes. We show that his conclusion holds under much milder assumptions, namely if the lattice polytope $P$ is spanning, i.e., any lattice point of the ambient lattice is an integer affine combination of lattice points in $P$. As an application, we get a generalization of Hibi's Lower Bound Theorem. Our proof relies on generalizing Bertini's theorem to the semistandard situation and Harris' Uniform Position Principle to certain curves in weighted projective space.