Generalizations of finite metrics and cuts (Q2813031)

This book is divided into five sections. The first three sections are largely a compendium of definitions, examples, and basic results about metrics (mainly on finite sets) and their generalizations and special cases. There are many such: quasimetrics, m-metrics, semimetrics, hemimetrics, cuts, hypermetrics, and more. The m-metrics and their various generalizations are of particular interest: these are functions \(X^{m+1} \rightarrow \mathbb R^+\) obeying metric-like axioms. A simple example is the area of the triangle defined by three points in the plane. Like many such measures, this can be 0 even when the three points are distinct.NEWLINENEWLINEThis first half of the book is extremely complete, and the researcher looking for an entry point into this area of mathematics will find it a reasonable reference. Most of the theorems proved in this part of the book are rather simple, the basic tools of the trade laid out for the newcomer. Regrettably, the book has no index, reducing its utility to the casual reader. The preview of the book in Chapter 1 partially fills this gap, but not adequately.NEWLINENEWLINEA somewhat picky point: there are many examples given of structures defined by axiom sets. These would be easier to read if, rather than listing the entire axiom set in each case, existing definitions were leveraged. Half a page of definitions (on page 84) could have been replaced by something like: ``A weak partial semimetric that obeys the \textit{small self-distance condition} \(p(x,y) \geq p(x,x)\) is called a \textit{partial semimetric}''. If it also obeys the \textit{strong self-distance condition} \(p(x,x)+p(y,y) \geq p(x,y)\), it is a \textit{strong partial semimetric.} (On page 85, in Definition 6.4, the authors do something like this, but confusingly exchange the new and old structures: they seem to define a partial semimetric to be a partial metric obeying the separation axiom, rather than the other way around.)NEWLINENEWLINEThe final two sections deal with the combinatorial geometry of finite metrics (and their various finite generalizations.) A metric on \(X\) being a function \(X^2 \rightarrow\mathbb R^+\), it may also be considered as a single point in the non-negative orthant of the vector space \(\mathbb R^{X\times X}\). But not all such functions are metrics; as the metric axioms are given by equalities and inequalities, the set of all metrics forms a cone within \(\mathbb R^{X\times X}\), an object of study in its own right. Many other metric-like structures obey similar axioms and likewise yield cones or polytopes. The main thrust of these sections is the description of these convex structures for small finite \(X\). While numerous useful theorems are given, the calculation rapidly becomes horrendous, and the main results (probably necessarily) report on the outcome of computer searches.NEWLINENEWLINEThe authors' English is generally clear but not absolutely idiomatic. A particularly common error is ``it holds (formula)'' for ``we have (formula).'' The authors are not to be blamed for this, but the publisher should have hired a line editor to correct such minor errors, which are ubiquitous.NEWLINENEWLINEThis book would be of interest to a specialist in the geometry of cuts and metrics, or to a researcher needing to browse the many options for generalizing the concept of ``metric''. The first group know who they are; the second group would be well served by the presence of a copy of this book in any university library.