A Blichfeldt-type inequality for the surface area

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\Z^n)\leq n! \vol(K)+n, whenever $K\subset {\mathbb{R}}^n$ is a convex body containing $n+1$ affinely independent integral points. Here we prove an analogous inequality with respect to the surface area $\text{F}(K)$, namely #(K\cap\Z^n) < \vol(K) + ((\sqrt{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 $t\in {\mathbb{R}}^n\setminus {\mathbb{Z}}^n$ and $P$ is an $n$-dimensional polytope with integral vertices. Then we have #((t+P)\cap\Z^n)\leq n! \vol(P). Moreover, in the 3-dimensional case we prove a stronger inequality, namely #(K\cap\Z^n) < \vol(K) + 2 \F(K).