Mathematika

Research Article

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

Professor Konrad J. Swanepoela1

a1 Department of Mathematical Sciences, University of South Africa, PO Box 392, Pretoria 0003, South Africa. E-mail: swanekj@unisa.ac.za

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.

(Received November 16 2004)

MSC (2000)

  • Primary, 52A37;
  • Secondary, 05C05, 49Q10, 52A21, 52A40, 52C17

Footnotes

Dedicated to Professor Rolf Schneider on the occasion of his 65th birthday