Lattice polytopes having -polynomials with given degree and linear coefficient

The h ∗ -polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let P be a lattice polytope with h ∗ -polynomial of degree d and with linear coefficient h 1 ∗ . We show that P has to be a lattice pyramid over a lower-dimensional lattice polytope if the dimension of P is greater than or equal to h 1 ∗ ( 2 d + 1 ) + 4 d − 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 ∗ -polynomial.