A unified simple proof of a conjecture of Woods for n⩽6

Let Rn be the n-dimensional Euclidean space. Let L denote a lattice in Rn of determinant 1 such that there is a sphere centered at the origin 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 contains a point of L. This is known to be true for n⩽6. Here we give a unified simple proof for n⩽6 of the more general conjecture of Woods.