Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees
| dc.creator | Swanepoel, Konrad J | |
| dc.date | 2004-10-06 | |
| dc.date.accessioned | 2026-07-07T07:48:45Z | |
| dc.date.available | 2026-07-07T07:48:45Z | |
| dc.description | In this note we prove two results on the quantitative illumination parameter f(d) of the unit ball of a d-dimensional normed space introduced by K. Bezdek (1992). The first is that f(d) = O(2^d d^2 log d). The second involves Steiner minimal trees. Let v(d) be the maximum degree of a vertex, and s(d) of a Steiner point, in a Steiner minimal tree in a d-dimensional normed space, where both maxima are over all norms. F. Morgan (1992) conjectured that s(d) <= 2^d, and D. Cieslik (1990) conjectured v(d) <= 2(2^d-1). We prove that s(d) <= v(d) <= f(d) which, combined with the above estimate of f(d), improves the previously best known upper bound v(d) < 3^d. | |
| dc.description | 5 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/math/0410144 | |
| dc.identifier | http://arxiv.org/abs/math/0410144 | |
| dc.identifier | Mathematika 52 (2005), 47--52 (2006). | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/124608 | |
| dc.subject | Metric Geometry | |
| dc.subject | 52A37 (Primary) 52A20, 52A40, 52C17 (Secondary) | |
| dc.title | Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees | |
| dc.type | text |