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The h^*-polynomial of a lattice polytope is the numerator of the generatingfunction of the Ehrhart polynomial. Let P be a lattice polytope withh^*-polynomial of degree d and with linear coefficient h^*_1. We show that Phas to be a lattice pyramid over a lower-dimensional lattice polytope, if thedimension of P is greater or equal to h^*_1 (2d+1) + 4d-1. This result has apurely combinatorial proof and generalizes a recent theorem of Batyrev.

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