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 $\#((t+P)\cap{\Bbb Z}^n)\le n! {\rm vol}(P)$.

Moreover, in the 3-dimensional case we prove a stronger inequality, namely $\#(K\cap{\Bbb Z}^n)< {\rm vol}(K) + 2 {\rm F}(K)$.