A Blichfeldt-type inequality for the surface area

Abstract

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 \(\#(K\cap{\Bbb Z}^n)\le n! {\rm vol}(K)+n\), whenever \(K\subset{\Bbb R}^n\) 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 \(\#(K\cap{\Bbb Z}^n) < {\rm vol}(K) + ((\sqrt{n}+1)/2) (n-1)! {\rm F}(K)\). 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 \(t\in{\Bbb R}^n\setminus{\Bbb Z}^n\) and P is an n-dimensional polytope with integral vertices. Then we have \(\#((t+P)\cap{\Bbb Z}^n)\le n! {\rm vol}(P)\).