Equality case in van der Corput's inequality and collisions in multiple lattice tilings

Van der Corput's provides the sharp bound vol(C) \le m 2^d on the volume of a d-dimensional origin-symmetric convex body C that has 2m-1 points of the integer lattice in its interior. For m=1, a characterization of the equality case vol(C)= m 2^d is equivalent to the well-known problem of characterizing tilings by translations of a convex body. It is rather surprising that so far, for m \ge 2, no characterization of the equality case has been available, though a hint to the respective characterization problem can be found in the 1987 monograph of Gruber and Lekkerkerker. We give an explicit characterization of the equality case for all m \ge 2. Our result reveals that, the equality case for m \ge 2 is more restrictive than for $m=1$. We also present consequences of our characterization in the context of multiple lattice tilings.