  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.
