Asymptotic formulae for the lattice point enumerator

Let M be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hiawka for large lambd a the number of lattice points in lambda M is given by G(lambda M) = V(lamb da M) + O(lambda(d-1-epsilon(d))) for some positive epsilon(d). Here we giv e for general convex bodies the weaker estimate \G(lambda M) - V(lambda M)\ less than or equal to 1/2 S-Zd(M)lambda(d-1) o(lambda(d-1)) where S-Zd (M) denotes the lattice surface area of M. The term S-Zd (M) is optimal for all convex bodies and o(lambda(d-1)) cannot be improved in gene ral. We prove that the same estimate even holds if we allow small deformati ons of M. Further we deal with families {P-lambda} of convex bodies where the only co ndition is that the inradius tends to infinity. Here we have \G(P-lambda) - V(P-lambda)\ less than or equal to dV(P-lambda, K; 1) + o(S( P-lambda)) where the convex body K satisfies some simple condition, V(P-lambda, K; 1) is some mixed volume and S(P-lambda) is the surface area of P-lambda.