Abstract
The idea of Voronoĭ’s proof of his well-known criterion that a positive definite quadratic form is extreme if and only if it is eutactic and perfect, is as follows: Identify positive definite quadratic forms on 𝔼d with their coefficient vectors in \({\mathbb{E}^{\frac{1}{d}\left( {d + 1} \right)}}\). This translates certain problems on quadratic forms into more transparent geometric problems in \({\mathbb{E}^{\frac{1}{d}\left( {d + 1} \right)}}\) which, sometimes, are easier to solve. Since the 1960s this idea has been applied successfully to various problems of quadratic forms, lattice packing and covering of balls, the Epstein zeta function, closed geodesics on the Riemannian manifolds of a Teichmüller space, and other problems.
This report deals with recent applications of Voronoĭ’s idea. It begins with geometric properties of the convex cone of positive definite quadratic forms and a finiteness theorem. Then we describe applications to lattice packings of balls and smooth convex bodies, to the Epstein zeta function and a generalization of it and, finally, to John type and minimum position problems.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Ash, A., On eutactic forms, Canad. J. Math., 29 (1977), 1040–1054.
Ash, A., On the existence of eutactic forms, Bull. London Math. Soc., 12 (1980), 192–196.
Bachoc, C., Designs, groups and lattices, J. Théor. Nombres Bordeaux, 17 (2005), 25–44.
Bachoc, C. and Venkov, B., Modular forms, lattices and spherical designs. Réseaux euclidiens, designs sphériques et formes modulaires, 87–111, Monogr. Enseign. Math. 37, Enseignement Math., Geneva, 2001.
Ball, K. M., Ellipsoids of maximal volume in convex bodies, Geom. Dedicata, 41 (1992), 241–250.
Bambah, R. P., On lattice coverings by spheres, Proc. Nat. Inst. Sci. India, A 23(954), 25–52.
Barnes, E. S. and Dickson, T. J., Extreme coverings of n-space by spheres, J. Austral. Math. Soc., 7 (1967), 115–127.
Barvinok, A., A course in convexity, Amer. Math. Soc., Providence, RI, 2002.
Bastero, J. and Romance, M., John’s decomposition of the identity in the non-convex case, Positivity, 6 (2002), 1–16.
Bavard, C., Systole et invariant d’Hermite, J. Reine Angew. Math., 482 (1997), 93–120.
Bavard, C., Théorie de Voronoĭ géométrique. Propriétés de finitude pour les familles de réseaux et analogues, Bull. Soc. Math. France, 133 (2005), 205–257.
Bergé, A.-M. and Martinet, J., Sur un probléme de dualité lié aux sphéres en géométrie des nombres, J. Number Theory, 32 (1989), 14–42.
Bergé, A.-M. and Martinet, J., Sur la classification des réseaux eutactiques, J. London Math. Soc. (2), 53 (1996), 417–432.
Bertraneu, A. and Fichet, B., Étude de la frontière de l’ensemble des formes quadratiques positives, sur un espace vectoriel de dimension finie, J. Math. Pures Appl. (9), 61 (1982), 207–218.
Böröczky, K., Jr. and Schneider, R., A characterization of the duality mapping for convex bodies, Geom. Funct. Anal., 18 (2008), 657–667.
Cassels, J. W. S., On a problem of Rankin about the Epstein zeta-function, Proc. Glasgow Math. Assoc., 4 (1959), 73–80.
Cohn, H. and Kumar, A., The densest lattice in twenty-four dimensions, Electron. Res. Announc. Amer. Math. Soc., 10 (2004), 58–67 (electronic).
Conway, J. H. and Sloane, N. J. A., Sphere packings, lattices and groups, 3rd ed. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov, Grundlehren Math. Wiss. 290, Springer-Verlag, New York, 1999.
Coulangeon, R., Spherical designs and zeta functions of lattices, Int. Math. Res. Not. Art. ID, 49620 (2006), 16pp.
Danzer, L., Laugwitz, D. and Lenz, H., Über das Löwnersche Ellipsoid und sein Analogon unter den einem Eikörper einbeschriebenen Ellipsoiden, Arch. Math., 8 (1957), 214–219.
Delone, B. N. and Ryshkov, S. S., A contribution to the theory of the extrema of a multi-dimensional ζ-function, Dokl. Akad. Nauk SSSR, 173 (1967), 991–994, Soviet Math. Dokl., 8 (1967), 499–503.
Delone, B. N., Dolbilin, N. P., Ryshkov, S. S. and Shtogrin, M. I., A new construction of the theory of lattice coverings of an n-dimensional space by congruent balls, Izv. Akad. Nauk SSSR Ser. Mat., 34 (1970), 289–298.
Engel, P., Geometric crystallography, in: Handbook of convex geometry, B 989–1041, North-Holland, Amsterdam, 1993.
Ennola, V., On a problem about the Epstein zeta-function, Proc. Cambridge Philos. Soc., 60 (1964), 855–875.
Erdös, P., Gruber, P. M. and Hammer, J., Lattice points, Longman Scientific, Harlow, Essex, 1989.
Giannopoulos, A. A. and Milman, V. D., Extremal problems and isotropic positions of convex bodies, Israel J. Math., 117 (2000), 29–60.
Giannopoulos, A. A. and Milman, V. D., Euclidean structure in finite dimensional normed spaces, in: Handbook of the geometry of Banach spaces, I 707–779, North-Holland, Amsterdam, 2001.
Giannopoulos, A. A. and Papadimitrakis, M., Isotropic surface area measures, Mathematika, 46 (1999), 1–13.
Giannopoulos, A. A., Perissinaki, I. and Tsolomitis, A., John’s theorem for an arbitrary pair of convex bodies, Geom. Dedicata, 84 (2001), 63–79.
Goethals, J.-M. and Seidel, J. J., Spherical designs, in: Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Columbus 1978) 255–272, Proc. Sympos. Pure Math. XXXIV, Amer. Math. Soc., Providence RI, 1979.
Gordon, Y., Litvak, A. E., Meyer, M. and Pajor, A., John’s decomposition in the general case and applications, J. Differential Geom., 68 (2004), 99–119.
Gruber, P. M., Die meisten konvexen Körper sind glatt, aber nicht zu glatt, Math. Ann., 228 (1977), 239–246.
Gruber, P. M., Typical convex bodies have surprisingly few neighbours in densest lattice packings, Studia Sci. Math. Hungar., 21 (1986), 163–173.
Gruber, P. M., Minimal ellipsoids and their duals, Rend. Circ. Mat. Palermo (2) 37 (1988), 35–64.
Gruber, P. M., The space of convex bodies, in: Handbook of convex geometry A, 301–318, North-Holland, Amsterdam, 1993.
Gruber, P. M., Baire categories in convexity, in: Handbook of convex geometry B, 1327–1346, North-Holland, Amsterdam, 1993.
Gruber, P. M., Convex and discrete geometry, Grundlehren Math. Wiss., 336, Springer, Berlin, Heidelberg, New York, 2007.
Gruber, P. M., Application of an idea of Voronoĭ to John type problems, Adv. in Math., 218 (2008), 299–351.
Gruber, P. M., Geometry of the cone of positive quadratic forms, Forum Math., 21 (2009), 147–166.
Gruber, P. M., On the uniqueness of lattice packings and coverings of extreme density, Adv. in Geom., 11 (2011), 691–710.
Gruber, P. M., Voronoĭ type criteria for lattice coverings with balls, Acta Arith., 149 (2011), 371–381.
Gruber, P. M., John and Löwner ellipsoids, Discrete Comput. Geom., 46 (2011), 776–788.
Gruber, P. M., Lattice packing and covering of convex bodies, Proc. Steklov Inst. Math., 275 (2011), 229–238.
Gruber, P. M., Application of an idea of Voronoĭ to lattice packing, in preparation.
Gruber, P. M., Application of an idea of Voronoĭ to lattice zeta functions, Proc. Steklov Inst. Math., 276 (2012), to appear.
Gruber, P. M. and Lekkerkerker, C. G., Geometry of numbers, 2nd ed., North-Holland, Amsterdam, 1987, Nauka, Moscow, 2008.
Gruber, P. M. and Schuster, F. E., An arithmetic proof of John’s ellipsoid theorem, Arch. Math. (Basel), 85 (2005), 82–88.
Grünbaum, B., Convex polytopes, 2nd ed., Prepared by V. Kaibel, V. Klee, G. M. Ziegler, Springer, New York, 2003.
John, F., Extremum problems with inequalities as subsidiary conditions, in: Studies and Essays, Presented to R. Courant on his 60th Birthday, January 8, 1948, 187–204, Interscience, New York, 1948.
Klee, V., Some new results on smoothness and rotundity in normed linear spaces, Math. Ann., 139 (1959), 51–63.
Klein, F., Die allgemeine lineare Transformation der Linienkoordinaten, Math. Ann., 2 (1870), 366–370, Ges. Math. Abh. I, Springer, Berlin, 1921.
Lim, S. C. and Teo, L. P., On the minima and convexity of Epstein zeta function, J. Math. Phys., 49 (2008), 073513, 25pp.
Lindenstrauss, J. and Milman, V. D., The local theory of normed spaces and its applications to convexity, in: Handbook of convex geometry B, 1154–1220, North-Holland, Amsterdam, 1993.
Martinet, J., Perfect lattices in Euclidean spaces, Grundlehren Math. Wiss. 325, Springer, Berlin, Heidelberg, New York, 2003.
Milman, V. D. and Pajor, A., Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Geometric aspects of functional analysis (1987–88) 64–104, Lecture Notes in Math., 1376, Springer, Berlin, 1989.
Montgomery, H. L., Minimal theta functions, Glasgow Math. J., 30 (1988), 75–85.
Pełczyński, A., Remarks on John’s theorem on the ellipsoid of maximal volume inscribed into a convex symmetric body in R n, Note Mat., 10 (1990), 395–410.
Plücker, J., Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raumelement, I. Abth. 1868 mit einem Vorwort von A. Clebsch., II. Abth. 1869, herausgegeben von F. Klein, Teubner, Leipzig, 1868.
Praetorius, D., Ellipsoide in der Theorie der Banachräume, Master Thesis, U Kiel, 2000.
Praetorius, D., Remarks and examples concerning distance ellipsoids, Colloq. Math., 93 (2002), 41–53.
Rankin, R. A., A minimum problem for the Epstein zeta-function, Proc. Glasgow Math. Assoc., 1 (1953), 149–158.
Rudelson, M., Contact points of convex bodies, Israel J. Math., 101 (1997), 93–124.
Ryshkov, S. S., On the question of the final ζ-optimality of lattices that yield the densest packing of n-dimensional balls, Sibirsk. Mat. Zh., 14 (1973), 1065–1075, 1158.
Ryshkov, S. S., Geometry of positive quadratic forms, in: Proc. Int. Congr. of Math. (Vancouver, 1974) 1, 501–506, Canad. Math. Congress, Montreal, 1975.
Ryshkov, S. S., On the theory of the cone of positivity and the theory of the perfect polyhedra Π(n) and n(m), Chebyshevskii Sb., 3 (2002), 84–96.
Ryshkov S. S. and Baranovskiĭ, E. P., Classical methods of the theory of lattice packings, Uspekhi Mat. Nauk, 34 (1979), 3–63, 236, Russian Math. Surveys, 34 (1979), 1–68.
Sarnak, P. and Strömbergsson, A., Minima of Epstein’s zeta function and heights of flat tori, Invent. Math., 165 (2006), 115–151.
Schmutz, P., Riemann surfaces with shortest geodesic of maximal length, Geom. Funct. Anal., 3 (1993), 564–631.
Schmutz Schaller (Schmutz), P., Systoles on Riemann surfaces, Manuscripta Math., 85 (1994), 428–447.
Schmutz Schaller, P., Geometry of Riemann surfaces based on closed geodesics, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 193–214.
Schmutz Schaller, P., Perfect non-extremal Riemann surfaces, Canad. Math. Bull., 43 (2000), 115–125.
Schneider, R., Convex bodies: the Brunn-Minkowski theory, Cambridge Univ. Press, Cambridge, 1993.
Schneider, R., The endomorphisms of the lattice of closed convex cones, Beiträge Algebra Geom., 49 (2008), 541–547.
Schürmann, A., Computational geometry of positive definite quadratic forms, Amer. Math. Soc., Providence, RI, 2009.
Schröcker, H.-P., Uniqueness results for minimal enclosing ellipsoids, Comput. Aided Geom. Design, 25 (2008), 756–762.
Schürmann, A. and F. Vallentin, F., Computational approaches to lattice packing and covering problems, Discrete Comput. Geom., 35 (2006), 73–116.
Swinnerton-Dyer, H. P. F., Extremal lattices of convex bodies, Proc. Cambridge Philos. Soc., 49 (1953), 161–162.
Venkov, B., Réseaux et designs sphériques, in: Réseaux euclidiens, designs sphériques et formes modulaires 10–86, Monogr. Enseign. Math., 37, Enseignement Math., Geneva, 2001.
Voronoĭ (Voronoï, Woronoi), G. F., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Première mémoire: Sur quelques propriétés des formes quadratiques positives parfaites, J. Reine Angew. Math., 133 (1908), 97–178, Coll. Works, II, 171–238.
Voronoĭ, G. F., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les paralléloèdres primitifs I, II, J. Reine Angew. Math., 134 (1908), 198–267 et 136 (1909), 67–181, Coll. Works, II, 239–368.
Voronoĭ, G. F., Collected works I➁III, Izdat. Akad. Nauk Ukrain. SSSR, Kiev, 1952.
Wickelgren, K., Linear transformations preserving the Voronoi polyhedron, Manuscript, 2001.
Zong, C., Sphere packings, Springer, New York, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
In memoriam László Fejes Tóth (1915–2005)
Rights and permissions
Copyright information
© 2013 János Bolyai Mathematical Society and Springer-Verlag
About this chapter
Cite this chapter
Gruber, P.M. (2013). Applications of an Idea of Voronoĭ, a Report. In: Bárány, I., Böröczky, K.J., Tóth, G.F., Pach, J. (eds) Geometry — Intuitive, Discrete, and Convex. Bolyai Society Mathematical Studies, vol 24. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41498-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-41498-5_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41497-8
Online ISBN: 978-3-642-41498-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)