2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/158436We consider Voronoi's reduction theory of positive definite quadratic forms which is based on Delone subdivision. We extend it to forms and Delone subdivisions having a prescribed symmetry group. Even more general, the theory is developed for forms which are restricted to a linear subspace in the space of quadratic forms. We apply the new theory to complete the classification of totally real thin algebraic number fields which was recently initiated by Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known sphere coverings in dimensions 9,..., 15.31 pages, 2 figures, 2 tables, (v4) minor changes, to appear in Duke Math. JMetric GeometryNumber Theory11H55, 52C17text