Unifying LLL inequalities

The Lenstra, Lenstra, and Lovász (abbreviated as LLL) basis reduction algorithm computes a basis of a lattice consisting of short, and near orthogonal vectors. The quality of an LLL reduced basis is expressed by three fundamental inequalities, and it is natural to ask, whether these have a common generalization. In this note we find unifying inequalities. Our main result is Theorem 1. Let b1, . . . , bn ∈ R be an LLL-reduced basis of the lattice L, 1 ≤ k ≤ j ≤ n, and d1, . . . , dj arbitrary linearly independent vectors in L. Then detL(b1, . . . , bk) ≤ 2(detL(d1, . . . , dj)) , (1) ‖b1 ‖ · · · ‖bk ‖ ≤ 2(detL(d1, . . . , dj)) . (2) By setting k and j to either 1 or n, from (1) we can recover the first two LLL inequalities, and from (2) we can recover all three. Even with one degree of freedom left, i.e. with k or j fixed to 1 or n, or k = j, we obtain generalizations that seem to be new. Our main lemma also generalizes a result of Lenstra, Lenstra and Lovász, and we believe that it is of independent interest: Lemma 1. Let d1, . . . , dk be linearly independent vectors from the lattice L, and b ∗ 1, . . . , b ∗ n the Gram Schmidt orthogonalization of an arbitrary basis. Then detL(d1, . . . , dk) ≥ min 1≤i1<···<ik≤n { ‖b∗i1 ‖ . . . ‖b ∗ ik ‖ }