Eutactic lattice

In mathematics, a eutactic lattice (or eutactic form) is a lattice in Euclidean space whose minimal vectors form a eutactic star. This means they have a set of positive eutactic coefficients c_i such that (v, v) = Σc_i(v, m_i)² where the sum is over the minimal vectors m_i. "Eutactic" is derived from the Greek language, and means "well-situated" or "well-arranged".

Voronoi (1908) proved that a lattice is extreme if and only if it is both perfect and eutactic.

Conway & Sloane (1988) summarize the properties of eutactic lattices of dimension up to 7.

References

• Conway, John Horton; Sloane, N. J. A. (1988), "Low-dimensional lattices. III. Perfect forms", Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences, 418 (1854): 43–80, doi:10.1098/rspa.1988.0073, ISSN 0962-8444, JSTOR 2398316, MR 953277
  • Conway, J. H.; Sloane, N. J. A. (1989), "Errata: Low-Dimensional Lattices. III. Perfect Forms", Proceedings of the Royal Society of London, 426 (1871): 441, doi:10.1098/rspa.1989.0134, JSTOR 2398351.
• Coxeter, Harold Scott MacDonald (1951), "Extreme forms", Canadian Journal of Mathematics, 3: 391–441, doi:10.4153/CJM-1951-045-8, ISSN 0008-414X, MR 0044580
• Korkine, A.; Zolotareff, G. (1877), "Sur les formes quadratique positives", Mathematische Annalen, 11 (2): 242–292, doi:10.1007/BF01442667, ISSN 0025-5831
• Martinet, Jacques (2003), Perfect lattices in Euclidean spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 327, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44236-3, MR 1957723
• Voronoi, G. (1908), "Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier Mémoire: Sur quelques propriétés des formes quadratiques positives parfaites", Journal für die reine und angewandte Mathematik (in French), 133 (133): 97–178, doi:10.1515/crll.1908.133.97, ISSN 0075-4102