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. In real Euclidean space R of dimension D there is the lattice Z of points with integer coordinates. Unless a different lattice is specified, a lattice point will mean a point of Z D , and a lattice simplex or lattice convex polytope will mean a simplex or convex polytope whose vertices are integer points, that is, elements of Z. The interior in R of a set S is denoted by S°; if the affine span of S has dimension less than D, we denote the relative interior of S by S'. Consider a lattice convex polytope P CR with the number K = #(P° Π Z) of interior lattice points non-zero, and with a total of J = $(P Π Z) lattice points. Our principal result is that J is bounded above by a function B(K, D) of K and D alone. For the case of zero symmetric convex polytopes P there is no need to assume that the vertices are lattice points. By Van der Corput's generalization of Minkowski's theorem vol(P) < K-2 [4].f By a theorem of Blichfeldt, if the lattice points of P span R, / < D + D\ vol(P) [I] 5 5 . Otherwise we can consider a subspace of R and get the same inequality / < D + D\K2. On the other hand if P need not be symmetric or have lattice point vertices then even for D — 2 and K — 1, / can be arbitrarily large. For instance, P might be the convex hull of (-«,0), (0,1 + \/n), («,0). With the restriction to lattice point vertices and D — 2 we have Scott's result that J < 3K + 7 (3K + 6 for K > 1), and of course when D = 1 we have trivially / < K + 2. These three bounds are best possible. Our results are far from best possible, but in any case the largest possible / grows rapidly with D, even for K — 1. Zaks, Perles and Wills have given examples of lattice simplices in R for which K = 1 and / > 2° ' [11]. There are some grounds for the belief that these examples are best possible. (See §4.) The existence of B(K, D) will follow from some facts about Diophantine approximation which we now establish.

Lattice vertex polytopes with interior lattice points