It was not clear to me what to expect from an encyclopedia of distances counting 650 pages. To me there was the mathematical definition of a distance, and I was wondering what the other 649.5 pages would contain. Well, by interpreting the word "distance" in its broadest possible sense, like two items being different, meaning that they are at a positive distance from each other, then it is difficult to imaging something not related to distance in this sense and you might wonder why this is not the encyclopedia of everything.
The authors however do attach a mathematical meaning to all these possible distances, meaning that there should be some mathematically well defined and computable quantification possible for what is measured. But even with this restriction, a distance can be defined in many contexts like all the subfields of classical mathematics (number theory, geometry, algebra, analysis, probability,...), applied mathematics, computer science, and social sciences. So the authors have grouped the 29 chapters into 7 parts, following more or less these general contexts. The different chapters then single out more specific subfields. For example distances on surfaces and knots, in functional analysis, in graph theory, in systems theory, in biology, ...
Now what does a chapter look like? It is mainly a bullet list of terms that are explained in a few lines. These terms are not only the surprisingly many different kinds of existing metrics, but these include also topics that are for example related to the spaces, graphs, networks etc. on which the distances are defined, or e.g. the surfaces and curves that are defined by distances, or the different meanings of similarity, and the likes of these. These few descriptive lines are mostly very compact, and hence it might require the introduction of many other definitions and concepts as well. For newer topics, or topics further away from mathematics in a stricter sense, there are less formulas and the entries are more verbose. I am certain though that for some items the reader needs to look up more details in the literature. For most (but not all) concepts the authors give (where possible) the original reference. That is the name of the author(s) and a year and the reader is supposed to be able to find the appropriate reference via the Internet. So it might not always be easy to find the intended literature and if it is found, the reference might not be accessible to the reader. In most instances, it will need more than just a few clicks to get it. Of course one does not need the original reference. Sometimes, a topic is better explained in a later reference work. Then of course there is a risk that it might be a generalization or some variation of what is mentioned in this book.
The book is clearly a reference work, i.e., a book in which to look up things. So for the paper version, the extensive index is essential. It is of course much more useful to have an online version where one may search for a word, and where keywords are referring to each other with internal or external links. That might have been possible in the eBook version. However that version is a collection of either a pdf or an html version of the chapters. The html version is however badly formatted, and more importantly, it contains only links to references that are listed at the end of the book, but no cross links or external links. Since readers are now used to wikipedia type of information, I believe this is a missed opportunity. Perhaps an idea for a next edition. The paper version is clearly a welcome asset in a mathematics library. Although many non-mathematical areas are covered as well, I think the approach is too mathematical to recommend it to other libraries. But in any case, libraries are at an increasing speed evolving towards web-based providers of their information, so also for them a true web-version would be most welcome.
The first edition (Springer, 2006) was an outgrow of their Dictionary of distances (Elsevier, 2006). This is the second edition in which corrections are implemented, together with new items and a bit streamlining, but the general structure is largely unaltered. Most changes are situated in sections on graph theory, engineering and the last part on "real world distances". Corrections to this edition will be maintained at www.liga.ens.fr/~deza/Distances.html