Motivated by mixed Hodge theory, Katz and Stapledon define refined versions of Ehrhart h^*-polynomials of lattice polytopes, one of them being the so-called local h^*-polynomial. We try to establish structural results on "thin" lattice polytopes with vanishing local h^*-polynomial. This is closely related to a still open problem in the seminal book by Gelfand, Kapranov and Zelevinsky. In another direction, we can also ask whether there is a generalization of the Katz--Stapledon apparatus for a family of lattice polytopes. This might potentially provide insight into two mysterious notions that recently appeared: the mixed degree of a family of lattice polytopes and the motivic geometric genus of its associated generic complete intersection.