  For any lattice polytope $P$, we consider an associated polynomial
$\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two
polynomials satisfying certain symmetry conditions. As a consequence, we
improve upon known inequalities satisfied by the coefficients of the Ehrhart
$\delta$-vector of a lattice polytope. We also provide combinatorial proofs of
two results of Stanley that were previously established using techniques from
commutative algebra. Finally, we give a necessary numerical criterion for the
existence of a regular unimodular lattice triangulation of the boundary of a
lattice polytope.
