Abstract Let R n be 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 R n 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 R 7 . Together with a result of C.T. McMullen (2005), the long standing conjecture of Minkowski follows for n = 7 .