Estimates on conjectures of Minkowski and woods II

Let ? ⁿ be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < Rwhich contains no point of ∧ other than the origin 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 ? ⁿ of radius \(\sqrt n /2\) contains a point of ∧. This is known to be true for n ≤ 8. Recently we gave estimates on a more general conjecture of Woods for n ≥ 9. This lead to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms. Here we shall refine our method to obtain improved estimates for Woods Conjecture. These give improved estimates of Minkowski’s conjecture for 9 ≤ n ≤ 31.