A unified simple proof of a conjecture of Woods for (original) (raw)

Let R n be the n-dimensional Euclidean space. Let L denote a lattice in R n 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 wellknown conjecture in the geometry of numbers asserts that any closed sphere in R n of radius 1 2 √ n 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.