Discrete & Computational Geometry

, Volume 5, Issue 4, pp 351–356 | Cite as

Diameter and radius in the Manhattan metric

  • D. Z. Du
  • D. J. Kleitman
Article

Abstract

We investigate maximum size sets of lattice points with a given diameter,d, within a given rectilinearly bounded finite regionR inn dimensions, under the Manhattan orL1 metric. We show that when the length ofR in each dimension is an odd integer (as, for example, then-cube) there is, for every integerd, a maximum size set having radiusd/2 about some center, though the center need not be a lattice point.

Similar results are obtained whenR has even length in some dimensions, except for a set ofd values whose cardinality is one less than the number of dimensions in whichR has even length. This question is still open for these values.

Keywords

Lattice Point Discrete Comput Geom Maximum Cardinality Complementary Pair Coordinate Difference 

References

  1. 1.
    D. J. Kleitman, On a combinatorial conjecture of Erdös,J. Combin. Theory 1 (1966) 209–214.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    D. J. Kleitman and M. Fellows, Radius and diameter in Manhattan lattices,Discrete Math., to appear.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • D. Z. Du
    • 1
  • D. J. Kleitman
    • 1
  1. 1.Mathematics DepartmentMassachusetts Institute of TechnologyCambridgeUSA

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