Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees

Swanepoel, Konrad ORCID: 0000-0002-1668-887X (2005) Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees. Mathematika, 52 (1). pp. 47-52. ISSN 0025-5793

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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.

Item Type:	Article
Official URL:	http://www.ucl.ac.uk/mathematics/Mathematika/
Additional Information:	© 2005 UCL
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	16 Oct 2009 09:42
Last Modified:	11 Dec 2024 22:54
URI:	http://eprints.lse.ac.uk/id/eprint/25456

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