On conjectures of minkowski and woods for n=7

Abstract

Let Rⁿbe the n-dimensional Euclidean space with O as the origin. Let ∧ be a lattice of determinant 1 such that there is a sphere |X|<R which contains no point of ∧ other than O and has n linearly independent points of ∧ on its boundary. A well-known conjecture in the geometry of numbers asserts that any closed sphere in Rⁿ of radius √n/4 contains a point of ∧. This is known to be true for n≤6. Here we prove a more general conjecture of Woods for n=7 from which this conjecture follows in R⁷. Together with a result of C.T. McMullen (2005), the long standing conjecture of Minkowski follows for n=7.