2007/05/15 by Martin Henk, J. M. Wills, Henk, Martin +1
Mathematics · #11H06 #52C07 #FOS: Mathematics #Mathematical Inequalities and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.0705.2088
openalex publication_date 2007/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1921 Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that #(K∩\Zn)≤ n! \vol(K)+n, whenever K⊂\Rn is a convex body containing n+1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area \F(K), namely #(K∩\Zn) < \vol(K) + ((√(n)+1)/2) (n-1)! \F(K). The proof is based on a slight improvement of Blichfeldt's bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K=t+P, where t∈\Rn∖\Zn and P is an n-dimensional polytope with integral vertices. Then we have #((t+P)∩\Zn)≤ n! \vol(P). Moreover, in the 3-dimensional case we prove a stronger inequality, namely #(K∩\Zn) < \vol(K) + 2 \F(K).