ar X iv : 0 70 6 . 41 78 v 1 [ m at h . C O ] 2 8 Ju n 20 07 Lattice polytopes of degree 2 Jaron Treutlein

Abstract. A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly i > 0 interior lattice points. We give a new proof for this theorem and classify polygons with maximal volume. Then we show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimension. From a theorem of Victor Batyrev the finiteness of lattice polytopes of degree 2 up to pyramid-constructions and affine unimodular transformations follows. Equivalently, there is only a finite number of quadratic polynomials with fixed leading coefficient being the h∗-polynomial of a lattice polytope.