2018/03/06 by Averkov, Gennadiy
#52C07 #52C22 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1803.02117
Van der Corput's provides the sharp bound vol(C) ≤ m 2d 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 2d 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 ≥ 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 ≥ 2. Our result reveals that, the equality case for m ≥ 2 is more restrictive than for m=1. We also present consequences of our characterization in the context of multiple lattice tilings.