Gitterpunktsanzahl und Parallelkörpervolumen von Eikörpern

Abstract

LetΛ={x=(x ₁,...,x _d )|x _i ∈ℤ} be a lattice in euclideand-spaceR ^d with respect to a rectangular coordinate system with unit vectors. Then for instance the following theorem holds: The number of lattice pointsG(K)=card (K∩Λ) of an arbitrary convex bodyK⊂R ^d is less or equal the volumeV(K _λd) of the outer parallel bodyK _λd ofK at a distance λ _d =ω ^-1/d _d (ω _d =Volume of thed-unitsphere, 2≤d≤5). The number λ is best possible.