In 1992 Du and Hwang published a paper confirming the correctness of a well
known 1968 conjecture of Gilbert and Pollak suggesting that the Euclidean
Steiner ratio for the plane is 2/3. The original objective of this thesis was to
adapt the technique used in this proof to obtain results for other Minkowski
spaces. In an attempt to create a rigorous and complete version of the proof,
some known results were given new proofs (results for hexagonal trees and
for the rectilinear Steiner ratio) and some new results were obtained (on
approximation of Steiner ratios and on transforming Steiner trees).
The most surprising result, however, was the discovery of a fundamental
gap in the proof of Du and Hwang. We give counter examples demonstrating
that a statement made about inner spanning trees, which plays an important
role in the proof, is not correct. There seems to be no simple way out of
this dilemma, and whether the Gilbert-Pollak conjecture is true or not for
any number of points seems once again to be an open question. Finally we
consider the question of whether Du and Hwang's strategy can be used for
cases where the number of points is restricted. After introducing some extra
lemmas, we are able to show that the Gilbert-Pollak conjecture is true for 7
or fewer points. This is an improvement on the 1991 proof for 6 points of
Rubinstein and Thomas. / Mathematical Sciences / Ph. D. (Mathematics)
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:unisa/oai:umkn-dsp01.int.unisa.ac.za:10500/1981 |
| Date | 31 January 2008 |
| Creators | De Wet, Pieter Oloff |
| Contributors | Swanepoel, K. J. |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 1 online resource (v, 75 leaves) |
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