  We prove several inequalities on the determinants of sublattices in
LLL-reduced bases. They generalize the inequalities on the length of the
shortest vector proven by Lenstra, Lenstra, and Lovasz, and show that
LLL-reduction finds not only a short vector, but more generally, sublattices
with small determinants. We also prove new upper bounds on the product of the
norms of the first few vectors.
