General Definitions

  • Michel Marie Deza
  • Elena Deza


Let X be a set. A function d:X×X→ℝ is called a distance (or dissimilarity) on X if, for all x,yX, there holds:
  1. 1.

    d(x,y)≥0 (nonnegativity);

  2. 2.

    d(x,y)=d(y,x) (symmetry);

  3. 3.

    d(x,x)=0 (reflexivity).



Chromatic Number Isometric Embedding Geodesic Segment Ultrametric Space Normed Vector Space 
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.


  1. [Amba76]
    Ambartzumian R. A Note on Pseudo-metrics on the Plane, Z. Wahrscheinlichkeitstheor. Verw. Geb., Vol. 37, pp. 145–155, 1976. MATHCrossRefMathSciNetGoogle Scholar
  2. [Badd92]
    Baddeley A.J. Errors in Binary Images and an L p Version of the Hausdorff Metric, Nieuw Arch. Wiskd., Vol. 10, pp. 157–183, 1992. MATHMathSciNetGoogle Scholar
  3. [BLMN05]
    Bartal Y., Linial N., Mendel M. and Naor A. Some Low Distortion Metric Ramsey Problems, Discrete Comput. Geom., Vol. 33, pp. 27–41, 2005. MATHCrossRefMathSciNetGoogle Scholar
  4. [Bloc99]
    Bloch I. On Fuzzy Distances and Their Use in Image Processing Under Unprecision, Pattern Recognit., Vol. 32, pp. 1873–1895, 1999. CrossRefGoogle Scholar
  5. [FoSc06]
    Foertsch T. and Schroeder V. Hyperbolicity, CAT(-1)-Spaces and the Ptolemy Inequality, arXiv:math.MG/0605418v2, 13 July 2006.
  6. [GoMc80]
    Godsil C.D. and McKay B.D. The Dimension of a Graph, Q. J. Math., Vol. 31, pp. 423–427, 1980. MATHCrossRefMathSciNetGoogle Scholar
  7. [Hemm02]
    Hemmerling A. Effective Metric Spaces and Representations of the Reals, Theor. Comput. Sci., Vol. 284-2, pp. 347–372, 2002. MATHCrossRefMathSciNetGoogle Scholar
  8. [Isbe64]
    Isbell J. Six Theorems About Metric Spaces, Comment. Math. Helv., Vol. 39, pp. 65–74, 1964. MATHCrossRefMathSciNetGoogle Scholar
  9. [Masc04]
    Mascioni V. Equilateral Triangles in Finite Metric Spaces, Electron. J. Comb., Vol. 11, R18, 2004. MathSciNetGoogle Scholar
  10. [Matt92]
    Matthews S.G. Partial Metric Topology, Research Report 212, Dept. of Computer Science, University of Warwick, 1992. Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michel Marie Deza
    • 1
  • Elena Deza
    • 2
  1. 1.École Normale SupérieureParisFrance
  2. 2.Moscow State Pedagogical UniversityMoscowRussia

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