Banach-Minkowski Planes

  • Dietmar Cieslik
Part of the Combinatorial Optimization book series (COOP, volume 10)

Abstract

In the former chapters we calculated or estimated the Steiner ratio for many metric spaces. We have seen that this quantity of a space lies between 0.5 and 1 and, moreover, these bounds are the best possible ones, even for finite-dimensional Banach spaces. In the present chapter we will see that the last fact is not true in two-dimensional Banach spaces. That means we will create better upper and lower bounds for the Steiner ratio of Banach-Minkowski planes.We find in theorem 7.4.1 that for any unit ball B in the plane the following is true:
$${m_2}\left( B \right)\frac{{\sqrt {13} - 1}}{3} = 0.8685...$$
A better bound is still unknown.

Keywords

Span Tree Unit Ball Full Tree Steiner Point Minkowski Plane 
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.

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Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Dietmar Cieslik
    • 1
  1. 1.Institute of Mathematics and C.S.University of GreifswaldGermany

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