Inhomogeneous extreme forms

Mathieu Dutour Sikirić[1]; Achill Schürmann[2]; Frank Vallentin[3]

  • [1] Rudjer Bosković Institute Bijenicka 54, 10000 Zagreb (Croatia)
  • [2] Universität Rostock Institut für Mathematik 18051 Rostock (Germany)
  • [3] Technical University of Delft Delft Institute of Applied Mathematics P.O. Box 5031, 2600 GA Delft (The Netherlands)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 6, page 2227-2255
  • ISSN: 0373-0956

Abstract

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G.F. Voronoi (1868–1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs.By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension  6 and prove their existence in all dimensions beyond.New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast with the packing problem.

How to cite

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Dutour Sikirić, Mathieu, Schürmann, Achill, and Vallentin, Frank. "Inhomogeneous extreme forms." Annales de l’institut Fourier 62.6 (2012): 2227-2255. <http://eudml.org/doc/251090>.

@article{DutourSikirić2012,
abstract = {G.F. Voronoi (1868–1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs.By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension $6$ and prove their existence in all dimensions beyond.New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast with the packing problem.},
affiliation = {Rudjer Bosković Institute Bijenicka 54, 10000 Zagreb (Croatia); Universität Rostock Institut für Mathematik 18051 Rostock (Germany); Technical University of Delft Delft Institute of Applied Mathematics P.O. Box 5031, 2600 GA Delft (The Netherlands)},
author = {Dutour Sikirić, Mathieu, Schürmann, Achill, Vallentin, Frank},
journal = {Annales de l’institut Fourier},
keywords = {lattices; Delone polytopes; spherical $t$-designs; sphere packing; sphere covering; Voronoi reduction theory; spherical -designs},
language = {eng},
number = {6},
pages = {2227-2255},
publisher = {Association des Annales de l’institut Fourier},
title = {Inhomogeneous extreme forms},
url = {http://eudml.org/doc/251090},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Dutour Sikirić, Mathieu
AU - Schürmann, Achill
AU - Vallentin, Frank
TI - Inhomogeneous extreme forms
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 6
SP - 2227
EP - 2255
AB - G.F. Voronoi (1868–1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs.By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension $6$ and prove their existence in all dimensions beyond.New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast with the packing problem.
LA - eng
KW - lattices; Delone polytopes; spherical $t$-designs; sphere packing; sphere covering; Voronoi reduction theory; spherical -designs
UR - http://eudml.org/doc/251090
ER -

References

top
  1. Avner Ash, On eutactic forms, Canad. J. Math. 29 (1977), 1040-1054 Zbl0339.52005MR491523
  2. E. S. Barnes, T. J. Dickson, Extreme coverings of n -space by spheres, J. Austral. Math. Soc. 7 (1967), 115-127 Zbl0164.35304MR215191
  3. Anne-Marie Bergé, Jacques Martinet, On weakly eutactic forms, J. Lond. Math. Soc. (2) 75 (2007), 187-198 Zbl1196.11099MR2302738
  4. H. F. Blichfeldt, The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z. 39 (1935), 1-15 Zbl0009.24403MR1545485
  5. Stephen Boyd, Lieven Vandenberghe, Convex optimization, (2004), Cambridge University Press, Cambridge Zbl1058.90049MR2061575
  6. Henry Cohn, Abhinav Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), 99-148 Zbl1198.52009MR2257398
  7. Henry Cohn, Abhinav Kumar, Optimality and uniqueness of the Leech lattice among lattices, Ann. of Math. (2) 170 (2009), 1003-1050 Zbl1213.11144MR2600869
  8. John H. Conway, Neil J. A. Sloane, The cell structures of certain lattices, Miscellanea mathematica (1991), 71-107, Springer, Berlin Zbl0738.52014MR1131118
  9. John H. Conway, Neil J. A. Sloane, Sphere packings, lattices and groups, 290 (1999), Springer-Verlag, New York Zbl0634.52002MR1662447
  10. H. S. M. Coxeter, Regular polytopes, (1973), Dover Publications Inc., New York MR370327
  11. B.N. Delone, N.P. Dolbilin, S.S. Ryshkov, M.I. Shtogrin, A new construction in the theory of lattice coverings of an n-dimensional space by equal spheres., Math. USSR, Izv. 4 (1970), 293-302 Zbl0217.18601
  12. P. Delsarte, J. M. Goethals, J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), 363-388 Zbl0376.05015MR485471
  13. Michel Marie Deza, Monique Laurent, Geometry of cuts and metrics, 15 (1997), Springer-Verlag, Berlin Zbl1210.52001MR1460488
  14. Mathieu Dutour Sikirić, Polyhedral package 
  15. Mathieu Dutour Sikirić, The six-dimensional Delaunay polytopes, European J. Combin. 25 (2004), 535-548 Zbl1046.52009MR2069380
  16. Mathieu Dutour Sikirić, Infinite series of extreme Delaunay polytope, European J. Combin. 26 (2005), 129-132 Zbl1062.52013MR2101040
  17. Mathieu Dutour Sikirić, Robert Erdahl, Konstantin Rybnikov, Perfect Delaunay polytopes in low dimensions, Integers 7 (2007) Zbl1194.52018MR2342197
  18. Mathieu Dutour Sikirić, Achill Schürmann, Frank Vallentin, A generalization of Voronoi’s reduction theory and its application, Duke Math. J. 142 (2008), 127-164 Zbl1186.11040MR2397885
  19. Mathieu Dutour Sikirić, Achill Schürmann, Frank Vallentin, Complexity and algorithms for computing Voronoi cells of lattices, Math. Comp. 78 (2009), 1713-1731 Zbl1215.11067MR2501071
  20. Robert M. Erdahl, A cone of inhomogeneous second-order polynomials, Discrete Comput. Geom. 8 (1992), 387-416 Zbl0773.11042MR1176378
  21. Robert M. Erdahl, Konstantin Rybnikov, On Voronoi’s two tilings of the cone of metrical forms, Rend. Circ. Mat. Palermo (2) Suppl. (2002), 279-296 Zbl1116.52006MR1962573
  22. Robert M. Erdahl, S. S. Ryshkov, The empty sphere, Canad. J. Math. 39 (1987), 794-824 Zbl0643.10025MR915016
  23. Wolfgang Lempken, Bernd Schröder, Pham Huu Tiep, Symmetric squares, spherical designs, and lattice minima, J. Algebra 240 (2001), 185-208 Zbl1012.11054MR1830550
  24. Jacques Martinet, Perfect lattices in Euclidean spaces, 327 (2003), Springer-Verlag, Berlin Zbl1017.11031MR1957723
  25. Curtis T. McMullen, Minkowski’s conjecture, well-rounded lattices and topological dimension, J. Amer. Math. Soc. 18 (2005), 711-734 (electronic) Zbl1132.11034MR2138142
  26. H. Minkowski, Diskontinuitätsbereich arithmetischer Äquivalenz, J. Reine Angew. Math. 129 (1905), 220-274 
  27. Marston Morse, Topologically non-degenerate functions on a compact n -manifold M ., J. Analyse Math. 7 (1959), 189-208 Zbl0096.30603MR113233
  28. Gabriele Nebe, Boris Venkov, Low-dimensional strongly perfect lattices. I. The 12-dimensional case, Enseign. Math. (2) 51 (2005), 129-163 Zbl1124.11031MR2154624
  29. J. Nottebaum, Sphärische 4-designs in Gittern, Universität Oldenburg (1995) 
  30. S. S. Ryshkov, R. M. Erdahl, The empty sphere. II, Canad. J. Math. 40 (1988), 1058-1073 Zbl0653.10027MR973509
  31. Alexander Schrijver, Theory of linear and integer programming, (1986), John Wiley & Sons Ltd., Chichester Zbl0970.90052MR874114
  32. Achill Schürmann, Computational geometry of positive definite quadratic forms, 48 (2009), American Mathematical Society, Providence, RI Zbl1185.52016MR2466406
  33. Achill Schürmann, Frank Vallentin, Local covering optimality of lattices: Leech lattice versus root lattice E 8 , Int. Math. Res. Not. (2005), 1937-1955 Zbl1156.11324MR2173600
  34. Achill Schürmann, Frank Vallentin, Computational approaches to lattice packing and covering problems, Discrete Comput. Geom. 35 (2006), 73-116 Zbl1091.52009MR2183491
  35. M. I. Štogrin, Locally quasidensest lattice packings of spheres, Dokl. Akad. Nauk SSSR 218 (1974), 62-65 Zbl0305.10023MR360476
  36. F. Vallentin, Sphere coverings, lattices, and tilings (in low dimensions), (2003) 
  37. B. B. Venkov, Réseaux euclidean, designs sphériques, et formes modulaires, Monogr. Enseign. Math. 37 (2001), 10-86. Zbl1139.11320MR1878745
  38. N. M. Vetčinkin, Uniqueness of classes of positive quadratic forms, on which values of Hermite constants are reached for 6 ≤ n ≤ 8 , Trudy Mat. Inst. Steklov. 152 (1980), 34-86, 237 Zbl0457.10013MR603814
  39. G. F. Voronoi, Nouvelles applications des paramètres continues à la théorie des formes quadratiques 1: Sur quelques propriétés des formes quadratiques positives parfaites, J. Reine Angew. Math. 133 (1908), 97-178 Zbl38.0261.01
  40. G. F. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxiéme Mémoire. Recherches sur les parallélloedres primitifs, J. Reine Angew. Math. 134 (1908), 198-287 

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