2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160476We 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.5 pagesNumber TheoryCombinatoricsMetric GeometryOptimization and Control11H06, 11H50, 11Y16text