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Quantitative Illumination of Convex Bodies and Vertex Degrees of Geometric Steiner Minimal Trees

Part of: Graph theory General convexity

Published online by Cambridge University Press:  21 December 2009

Konrad J. Swanepoel
Konrad J. Swanepoel
Affiliation:
Department of Mathematical Sciences, University of South Africa, PO Box 392, Pretoria 0003, South Africa. E-mail: swanekj@unisa.ac.za
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Abstract

Two results are proved involving the quantitative illumination parameter B(d) of the unit ball of a d-dimensional normed space introduced by Bezdek (1992). The first is that B(d) = O(2dd2 log d). The second involves Steiner minimal trees. Let v(d) be the maximum degree of a vertex, and s(d) that of a Steiner point, in a Steiner minimal tree in a d-dimensional normed space, where both maxima are over all norms. Morgan (1992) conjectured that s(d) ≤ 2d, and Cieslik (1990) conjectured that v(d) ≤ 2(2d − 1). It is proved that s(d) ≤ v(d) ≤ B(d) which, combined with the above estimate of B(d), improves the previously best known upper bound v(d) < 3d.

MSC classification

Secondary: 52A37: Other problems of combinatorial convexity 05C05: Trees

Information

Type
Research Article
Information
Mathematika , Volume 52 , Issue 1-2 , December 2005 , pp. 47 - 52
Copyright
Copyright © University College London 2005

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