Metric Spaces in Pure and Applied Mathematics

The close relationship between the theory of quadratic forms and distance analysis has been known for centuries, and the theory of metric spaces that formalizes distance analysis and was developed over the last century, has obvious strong relations to quadratic-form theory. In contrast, the first paper that studied metric spaces {\em as such} -- without trying to study their embeddability into any one of the standard metric spaces nor looking at them as mere `presentations' of the underlying topological space -- was, to our knowledge, written in the late sixties by John Isbell. In particular, Isbell showed that in the category whose objects are metric spaces and whose morphisms are {\em non-expansive} maps, a unique {\em injective hull} exists for every object, he provided an explicit construction of this hull, and he noted that, at least for finite spaces, it comes endowed with an intrinsic polytopal cell structure. In this paper, we discuss Isbell's construction, we summarize the history of --- and some basic questions studied in --- {\it phylogenetic analysis}, and we explain why and how these two topics are related to each other. Finally, we just mention in passing some intriguing analogies between, on the one hand, a certain stratification of the cone of all metrics defined on a finite set $X$ that is based on the combinatorial properties of the polytopal cell structure of Isbell's injective hulls and, on the other, various stratifications of the cone of positive semi-definite quadratic forms defined on ${\mathbb R}^n$ that were introduced by the Russian school in the context of reduction theory.

2000 Mathematics Subject Classification: 15A63, 05C05, 92-02, 92B99

Keywords and Phrases: injective hull, tight span, phylogenetic tree, quadratic form

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