Geometriae Dedicata

, Volume 69, Issue 3, pp 237–240 | Cite as

A Note on Circular Decomposable Metrics

  • Victor Chepoi
  • Bernard Fichet


A metric d on a finite set X is called a Kalmanson metric if there exists a circular ordering ζ of points of X, such that d(y, u) + d(z, v) ≥ d(y, z) + d(u, v) for all crossing pairs yu and zv of ζ. We prove that any Kalmanson metric d is an l1-metric, i.e. d can be written as a nonnegative linear combination of split metrics. The splits in the decomposition of d can be selected to form a circular system of splits in the sense of Bandelt and Dress.

l1-metric circular decomposable metric Crofton formula 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexander, R.: Planes for which the lines are the shortest paths between points, Illinois J. Math. 22 (1978), 177–190.Google Scholar
  2. 2.
    Bandelt, H.J. and Dress, A.W.M.: A canonical decomposition theory for metrics on a finite set, Adv. Math. 92 (1992), 47–105.Google Scholar
  3. 3.
    Bretagnolle, J., Dacunha Castelle, D., and Krivine, J.L.: Lois stables et espaces L p, Ann. Institute Henri Poincaré sect. B 2(3) (1966), 231–259.Google Scholar
  4. 4.
    Christopher, G., Farach, M., and Trick, M. A.: The structure of circular decomposable metrics, In: J. Diaz and M. Serna (eds), Algorithms-ESA'96, Lecture Notes in Computer Science 1136 Springer, New York, 1996, pp. 486–500.Google Scholar
  5. 5.
    Deineko, V., Rudolf, R. and Woeginger, G.: Sometimes traveling is easy: the master tour problem, Technical Report, Institute of Mathematics, University of Technology, Graz (1995).Google Scholar
  6. 6.
    Deza, M. and Laurent, M.: Geometry of Cuts and Metrics, Springer, Berlin, 1997.Google Scholar
  7. 7.
    Kalmanson, K., Edgeconvex circuits and the travelling salesman problem, Canadian J. Math. 27 (1975), 1000–1010.Google Scholar
  8. 8.
    Rinow, W.: Die innere Geometrie der metrischen Räume, Springer, Berlin, 1961.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Victor Chepoi
    • 1
  • Bernard Fichet
    • 1
  1. 1.Laboratoire de BiomathématiquesUniversité d'Aix Marseille IIMarseille Cedex 5France

Personalised recommendations