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Discrete & Computational Geometry

, Volume 21, Issue 3, pp 437–447 | Cite as

Vertex Degrees of Steiner Minimal Trees in p d and Other Smooth Minkowski Spaces

  • K. J. Swanepoel

Abstract.

We find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees (SMTs) in the d -dimensional Banach spaces \( \ell\) p d independent of d . This is in contrast to Minimal Spanning Trees, where the maximum degree of vertices grows exponentially in d [19]. Our upper bounds follow from characterizations of singularities of SMTs due to Lawlor and Morgan [14], which we extend, and certain \( \ell\) p -inequalities. We derive a general upper bound of d+1 for the degree of vertices of an SMT in an arbitrary smooth d -dimensional Banach space (i.e. Minkowski space); the same upper bound for Steiner points having been found by Lawlor and Morgan. We obtain a second upper bound for the degrees of vertices in terms of 1 -summing norms.

Keywords

Banach Space Span Tree Maximum Degree Minkowski Space Minimal Span Tree 
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.

Copyright information

© Springer-Verlag New York Inc. 1999

Authors and Affiliations

  • K. J. Swanepoel
    • 1
  1. 1.Department of Mathematics and Applied Mathematics, University of Pretoria, 0002 Pretoria, South Africa konrad@math.up.ac.zaZA

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