A Blichfeldt-type Inequality for the Surface Area
نویسندگان
چکیده
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∩Z) ≤ n! vol(K)+n, whenever K ⊂ R 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 ∩ Z) < 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 ∈ R \Z and P is an n-dimensional polytope with integral vertices. Then we have #((t + P ) ∩ Z) ≤ n! vol(P ). Moreover, in the 3-dimensional case we prove a stronger inequality, namely #(K ∩ Z) < vol(K) + 2F(K).
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