Successive minima,intrinsic volumes,and lattice determinants

In Euclideand-spaceE ^d we prove a lattice-point inequality for arbitrary lattices and for the intrinsic volumesV _i (i.e., normalized quermassintegrals) of convex bodies. TheV _i are not equi-affine invariant (except the volume), hence suitable functionals of the lattice have to be introduced. The result generalizes an earlier result of Henk for the integer lattice ℤ ^d .