Geometry of minimal networks and the one-dimensional Plateau problem

Abstract

CONTENTS Introduction §1. The Steiner problem and its variations 1.1. Fundamental definitions 1.2. Globally minimal networks 1.3. Locally minimal networks 1.4. Closed networks and networks with a fixed boundary 1.5. Local structure of minimal networks 1.6. Minimal networks in classical ambient spaces 1.6.1. The case of the two-dimensional Euclidean plane 1.6.2. The case of two-dimensional closed surfaces 1.6.3. The case of polyhedra 1.6.4. The case of three-dimensional Euclidean space §2. Globally minimal networks on the plane 2.1. Minimal spanning trees 2.2. Travelling salesperson problem 2.3. Minimal Steiner trees 2.3.1. Lune property 2.3.2. Wedge property 2.3.3. Double wedge property 2.3.4. Connection between the minimal Steiner tree and the EMST 2.3.5. Diamond property 2.3.6. Convex hull and Steiner hull 2.3.7. The minimal Steiner tree and the Simpson lines 2.4. Steiner ratio 2.5. Globally minimal networks spanning points lying on a “ladder” and a “zigzag line” 2.6. Globally minimal networks spanning points lying on a circle 2.7. Improved algorithm for finding a minimal Steiner tree §3. Locally minimal networks on the plane 3.1. Complete classification of minimal 2-trees with a convex boundary 3.1.1. Rotation number 3.1.2. Parquet realization of 2-trees with rotation number not exceeding five 3.1.3. Parquets and their properties 3.1.4. Classification theorems 3.2. Non-degenerate minimal networks with a convex boundary. Cyclic case 3.2.1. Trivial networks and their rotation numbers 3.2.2. Parquet realization of networks with a convex minimal realization 3.2.3. Description of parquets of networks with a convex minimal realization 3.3. Minimal 2-trees spanning the vertices of regular n-gons 3.4. Convex minimal realization of degenerate Steiner trees References