Ryshkov, S. S.; Bol’shakova, E. A. On the theory of mainstay parallelohedra. (English. Russian original) Zbl 1104.51012 Izv. Math. 69, No. 6, 1257-1277 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 6, 187-210 (2005). Summary: In [S. S. Ryshkov, “The structure of primitive parallelohedra and Voronoi’s last problem”, Usp. Mat. Nauk 53, No. 2. 161–162 (1998), Russ. Math. Surv. 53, No. 2, 403–405 (1998; Zbl 0927.51028)] the first author announced a theorem stating that every primitive \(n\)-dimensional parallelohedron can be represented, up to an affine transformation, as a weighted Minkowski sum of parallelohedra belonging to a certain finite set of \(n'\)-dimensional \((n'\leq n)\) mainstay parallelohedra situated in a special way. This paper contains a detailed proof of this theorem in a refined and definitive form. Cited in 4 Documents MSC: 51M20 Polyhedra and polytopes; regular figures, division of spaces 51H20 Topological geometries on manifolds 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 11H31 Lattice packing and covering (number-theoretic aspects) 52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry) 52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry) 11H06 Lattices and convex bodies (number-theoretic aspects) Citations:Zbl 0927.51028 PDFBibTeX XMLCite \textit{S. S. Ryshkov} and \textit{E. A. Bol'shakova}, Izv. Math. 69, No. 6, 1257--1277 (2005; Zbl 1104.51012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 6, 187--210 (2005) Full Text: DOI