Swanepoel, Konrad
(1999)
*Vertex degrees of Steiner minimal trees in ℓ p d and other smooth Minkowski spaces.*
Discrete and Computational Geometry, 21
(3).
pp. 437-447.
ISSN 0179-5376

## Abstract

We find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees (SMTs) in the d -dimensional Banach spaces 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 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.

Item Type: | Article |
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Official URL: | http://www.springer.com/math/numbers/journal/454 |

Additional Information: | © 2009 Springer |

Library of Congress subject classification: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Date Deposited: | 09 Oct 2009 09:47 |

URL: | http://eprints.lse.ac.uk/25410/ |

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