Generalizations of Finite Metrics and Cuts

This book introduces oriented version of metrics and cuts and their multidimensional analogues, as well as partial metrics and weighted metrics. It is a follow-up of Geometry of Cuts and Metrics by Deza and Laurent which presents rich theory of classical binary and symmetric objects — metrics and cuts.

Many research publications on this subject are devoted to different special aspects of the theory of generalized metrics. However, they are disconnected one from other, often written in different mathematical language, consider the same objects from different points of view without analysis of possible connections, etc. In this book we will construct full theory of main classes of finite generalized metrics and their polyhedral aspects.

Contents:

• Preliminaries:

  • Short Preview of the Book
  • Main Definitions

• Main Notions and Examples:

  • Non-Oriented Case: Metrics
  • Oriented Case: Quasi-Metrics
  • Multidimensional Case: m-Metrics
  • Important Special Case: Partial Metrics and Weightable Quasi-Metrics

• Cuts, Hypermetrics and Their Generalizations:

  • Cuts and Their Generalizations
  • Hypermetrics and Their Generalizations

• Cones and Polytopes of Generalized Finite Semimetrics:

  • Non-Oriented Case: Semimetrics and Cuts
  • Oriented Case: Quasi-Semimetrics and Oriented Cuts
  • Multidimensional Case: m-Hemimetrics

• Important Cases of Polyhedra of Generalized Finite Semimetrics:

  • Cones of Partial Semimetrics and Weightable Quasi-Semimetrics
  • Cones of Hypermetrics
  • Cuts Over General Graphs
  • Connections Between Generalized Metrics Polyhedra

Readership: Students and researchers in discrete geometry, discrete mathematics, computation mathematics, finite geometry and related fields, as well as general audience/amateurs of mathematics.