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 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.
Duke U.P.
Duke Mathematical Journal
Semidefinite programming and combinatorial optimization , Spinoza prijs Lex Schrijver
Networks and Optimization

Dutour Sikirić, M., Schuermann, A., & Vallentin, F. (2008). A generalization of Voronoi's reduction theory and its application. Duke Mathematical Journal, 142(1), 127–164.