Variational approach and Steiner minimal trees on four points

✍ Scribed by J.F. Weng

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 691 KB
• Volume: 132
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)90244-5

✦ Synopsis

A Steiner topology t, is defined to be a degenerate topology of another Steiner topology t, if t, is obtained from t, by sequentially collapsing Steiner points into their adjacent regular points of degree less than three. A full topology t is called optimal if there is a Steiner minimal tree whose topology is degenerated from t. Because for any full topology t there is at most one Steiner tree with a topology degenerated from t, the key to the Steiner problem is to find the optimal full topology. We use this idea and the variational approach to make a systematic study of the Steiner problem on four points. A quadrilateral

P[abcd]

(not necessarily convex) is called skew if either both labj>ladl and Icd)>lcbl or both (abl>JbC and IcdI>lad(; otherwise, the opposite of the shortest side is the longest side. In this paper we completely determine the optimal topology for skew quadrilaterals, and give some sufficient conditions of the optimal topology for not skew quadrilaterals.

We also prove a general imbedding theorem of optimal topologies.