Interpolating between volume and lattice point enumerator with successive minima

We study inequalities that simultaneously relate the number of lattice points, volume and successive minima a convex body to one another. One main ingredient in order establish these relations is Blaschke's shaking procedure, by which problem can be reduced from arbitrary bodies anti-blocking bodies. As consequence our results, we obtain an upper bound on point enumerator terms minima, equivalent Minkowski's minima.