Swanepoel, Konrad
(2005)
*Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees.*
Mathematika, 52 (1).
pp. 47-52.
ISSN 0025-5793

## 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 |
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Official URL: | http://www.ucl.ac.uk/mathematics/Mathematika/ |

Additional Information: | © 2005 UCL |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Date Deposited: | 16 Oct 2009 09:42 |

Last Modified: | 20 Apr 2019 00:33 |

URI: | http://eprints.lse.ac.uk/id/eprint/25456 |

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