We 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 generally, the theory is developed for forms that 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 [BF] and Bayer-Fluckiger and Nebe [BFN]. Moreover, we apply it to construct new best-known sphere coverings in dimensions .
"A generalization of Voronoi's reduction theory and its application." Duke Math. J. 142 (1) 127 - 164, 15 March 2008. https://doi.org/10.1215/00127094-2008-003