The hypermetric cone is polyhedral

Abstract

The hypermetric coneH n is the cone in the spaceR n(n−1)/2 of all vectorsd=(d ij)1≤i<jn satisfying the hypermetric inequalities: −1≤ijn z j z j d ij ≤ 0 for all integer vectorsz inZ n with −1≤in z i =1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres andL-polytopes in lattices). As an application, we show that the hypermetric coneH n is polyhedral.

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References

  1. [1]

    P. Assouad: Sous espaces deL 1 et inégalités hypermétriques,Comptes Rendus de l'Académie des Sciences de Paris 294 (1982), 439–442.

    Google Scholar 

  2. [2]

    P. Assouad: Sur les inégalités valides dansL 1,European Journal of Combinatorcs 5 (1984), 99–112.

    Google Scholar 

  3. [3]

    P. Assouad, andM. Deza: Espaces métriques plongeables dans un hypercube: aspects combinatoires,Annals of Discrete Mathematics 8 (1980),197–210.

    Google Scholar 

  4. [4]

    D. Avis: On the extreme rays of the metric cone,Canadian Journal of Mathematics 32(1) (1980), 126–144.

    Google Scholar 

  5. [5]

    D. Avis, andM. Deza:L 1-embeddability, complexity and multicommodity flows,Networks 21 (1991), 595–617.

    Google Scholar 

  6. [6]

    D. Avis, andMutt: All the facets of the six point Hamming cone,European Journal of Combinatorics 10 (1989), 309–312.

    Google Scholar 

  7. [7]

    E. P. Baranovskii: Simplexes ofL-decompositions of Euclidian spaces,Matematicheskie Zametki (in Russian)10(6) (1971), 659–670; English translation inMathematical notes 10 (1971), 827–834.

    Google Scholar 

  8. [8]

    J. W. S. Cassels:An introduction to the geometry of numbers, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 99, Springer Verlag, 1959.

  9. [9]

    J. H. Conway, andN. J. A. Sloane:Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften 290, Springer Verlag, 1987.

  10. [10]

    L. Danzer, andB. Grünbaum: Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V. L. Klee,Math. Zeitschrift 79 (1962), 95–99.

    Google Scholar 

  11. [11]

    M. Deza: On the Hamming geometry of unitary cubes,Doklady Akad. Nauk. SSSR (in Russian)134 (1960), 1037–1040; English translation inSoviet Physics Dokl.5 (1961), 940–943.

    Google Scholar 

  12. [12]

    M. Deza: Matrices de formes quadratiques non négatives pour des arguments binaires.Comptes Rendus de l'Académie des Sciences de Paris 277 (1973), 873–875.

    Google Scholar 

  13. [13]

    M. Deza, V. P. Grishukhin, andM. Laurent: Extreme hypermetrics andL-polytopes, in Sets, Graphs and Numbers, Budapest, 1991, volume 60 of Colloquia Mathematica Societatis János Bolyai (1992), 157–209.

  14. [14]

    M. Deza, andM. Laurent: Facets for the cut cone I,Mathematical Programming 56 (2) (1992), 121–160.

    Google Scholar 

  15. [15]

    M. Deza, andM. Laurent: Facets for the cut cone II: clique-web inequalities,Mathematical Programming 56 (2) (1992), 161–188.

    Google Scholar 

  16. [16]

    R. M. Erdahl andS. S. Ryshkov: The empty sphere,Canadian Journal of Mathematics 34 (1987), 794–824.

    Google Scholar 

  17. [17]

    V. P. Grishukhin: Computing extreme rays of the metric cone for seven points,European Journal of Combinatorics 13 (1992), 153–165.

    Google Scholar 

  18. [18]

    J. B. Kelly: Hypermetric spaces, in: Lecture Notes in Mathematics 490 (1974), 17–31.

  19. [19]

    P. Terwiliger, andM. Deza: Classification of finite connected hypermetric spaces,Graphs and Combinatorics 3 (1987), 293–298.

    Google Scholar 

  20. [20]

    G. F. Voronoi: Nouvelles applications des paramètres continus à la théorie des formes quadratiques, Deuzième mémoire.J. Reine Angew. Math. 134 (1908) 198–287,136 (1909), 67–178.

    Google Scholar 

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Deza, M., Grishukhin, V.P. & Laurent, M. The hypermetric cone is polyhedral. Combinatorica 13, 397–411 (1993). https://doi.org/10.1007/BF01303512

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AMS subject classification code (1991)

  • 52 A 43
  • 52 A 25
  • 05 A 99