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Vertex degrees of Steiner minimal trees in ℓ p d and other smooth Minkowski spaces

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

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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
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
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Date Deposited: 09 Oct 2009 09:47
URL: http://eprints.lse.ac.uk/25410/

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