Lattice polytopes having h∗h∗-polynomials with given degree and linear coefficient

The h∗h∗-polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let PP be a lattice polytope with h∗h∗-polynomial of degree dd and with linear coefficient h1∗. We show that PP has to be a lattice pyramid over a lower-dimensional lattice polytope if the dimension of PP is greater than or equal to h1∗(2d+1)+4d−1. This result generalizes a recent theorem of Batyrev. As an application we deduce from an inequality due to Stanley that the volume of a lattice polytope is bounded by a function depending only on the degree and the two highest non-zero coefficients of the h∗h∗-polynomial.