On a problem of Danzer.

By a Danzer set S we shall mean a subset of the ^-dimensional Euclidean space Rn which has the property that every closed convex body of volume one in Rn contains a point of S. L. Danzer has asked if for n ^ 2 there exist such sets S with a finite density. The answer to this question is still unknown. In this note our object is to prove two theorems about Danzer sets. If A is a w-dimensional lattice, any translate Γ = A + p of A will be called a grid Γ; A will be called the lattice of Γ and the determinant d(A) of A will be called the determinant of Γ and will be denoted by d(Γ). In § 2 we prove