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?Bbb Zⁿ) ￡ n! vol(K)+n#(Kcap{Bbb Z}^n)le n! {rm vol}(K)+n , whenever K ì Bbb RⁿKsubset{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?Bbb Zⁿ) < vol(K) + ((?n+1)/2) (n-1)! F(K)#(Kcap{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 ? Bbb RⁿBbb Zⁿtin{Bbb R}^nsetminus{Bbb Z}^n and P is an n-dimensional polytope with integral vertices. Then we have #((t+P)?Bbb Zⁿ) ￡ n! vol(P)#((t+P)cap{Bbb Z}^n)le n! {rm vol}(P) . Moreover, in the 3-dimensional case we prove a stronger inequality, namely #(K?Bbb Zⁿ) < vol(K) + 2 F(K)#(Kcap{Bbb Z}^n)< {rm vol}(K) + 2 {rm F}(K) .