Institutional Repository

Geometric Steiner minimal trees

Show simple item record

dc.contributor.advisor Swanepoel, K. J. en
dc.contributor.author De Wet, Pieter Oloff en
dc.date.accessioned 2009-08-25T10:58:41Z
dc.date.available 2009-08-25T10:58:41Z
dc.date.issued 2009-08-25T10:58:41Z
dc.date.submitted 2008-01-31 en
dc.identifier.citation De Wet, Pieter Oloff (2009) Geometric Steiner minimal trees, University of South Africa, Pretoria, <http://hdl.handle.net/10500/1981> en
dc.identifier.uri http://hdl.handle.net/10500/1981
dc.description.abstract 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. en
dc.format.extent 1 online resource (v, 75 leaves)
dc.language.iso en en
dc.subject Minkowski space en
dc.subject Spanning tree en
dc.subject Minimum spanning tree en
dc.subject Steiner tree en
dc.subject Steiner minimal tree en
dc.subject Steiner ratio en
dc.subject Rectilinear tree en
dc.subject Hexagonal tree en
dc.subject Surface en
dc.subject Inner spanning tree en
dc.subject Inner Steiner tree en
dc.subject.ddc 512.55
dc.subject.lcsh Steiner systems
dc.title Geometric Steiner minimal trees en
dc.type Thesis en
dc.description.department Mathematical Sciences en
dc.description.degree Ph. D. (Mathematics) en


Files in this item

This item appears in the following Collection(s)

Show simple item record

Search UnisaIR


Browse

My Account

Statistics