Covering six space with spheres

AbstractLet Rn be euclidean n-space with a cartesian coordinate system in which O is the origin. Let L denote a lattice in Rn of determinant 1 such that there is a sphere centred at O which contains n linearly independent points of L on its boundary but no point of L other than O inside it. A well-known conjecture in the geometry of numbers asserts that any closed sphere in Rn of radius 12√n contains a point of L. This is known to be true for n ≤ 5. Here we prove that it is true for n = 6