Geometry of minimal networks and the one-dimensional Plateau problem
Alexander Ivanov, Alexey Avgustinovich Tuzhilin · Russian Mathematical Surveys · 1992
CONTENTSIntroduction §1. The Steiner problem and its variations1.1. Fundamental definitions1.2. Globally minimal networks1.3. Locally minimal networks1.4. Closed networks and networks with a fixed boundary1.5. Local structure of minimal networks1.6. Minimal networks in classical ambient spaces1.6.1. The case of the two-dimensional Euclidean plane1.6.2. The case of two-dimensional closed surfaces1.6.3. The case of polyhedra1.6.4. The case of three-dimensional Euclidean space §2. Globally minimal networks on the plane2.1. Minimal spanning trees2.2. Travelling salesperson problem2.3. Minimal Steiner trees2.3.1. Lune property2.3.2. Wedge property2.3.3. Double wedge property2.3.4. Connection between the minimal Steiner tree and the EMST2.3.5. Diamond property2.3.6. Convex hull and Steiner hull2.3.7. The minimal Steiner tree and the Simpson lines2.4. Steiner ratio2.5. Globally minimal networks spanning points lying on a ladder and a zigzag line2.6. Globally minimal networks spanning points lying on a circle2.7. Improved algorithm for finding a minimal Steiner tree §3. Locally minimal networks on the plane3.1. Complete classification of minimal 2-trees with a convex boundary3.1.1. Rotation number3.1.2. Parquet realization of 2-trees with rotation number not exceeding five3.1.3. Parquets and their properties3.1.4. Classification theorems3.2. Non-degenerate minimal networks with a convex boundary. Cyclic case3.2.1. Trivial networks and their rotation numbers3.2.2. Parquet realization of networks with a convex minimal realization3.2.3. Description of parquets of networks with a convex minimal realization3.3. Minimal 2-trees spanning the vertices of regular n-gons3.4. Convex minimal realization of degenerate Steiner treesReferences