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Metric Spaces in Pure and Applied Mathematics Documenta Math. 121 Metric Spaces in Pure and Applied Mathematics A. Dress, K. T. Huber1, V. Moulton2 Received: September 25, 2001 Revised: November 5, 2001 Communicated by Ulf Rehmann Abstract. The close relationship between the theory of quadratic forms and distance analysis has been known for centuries, and the the- ory of metric spaces that formalizes distance analysis and was devel- oped over the last century, has obvious strong relations to quadratic- form theory. In contrast, the ¯rst paper that studied metric spaces 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 non-expansive maps, a unique injective hull exists for every object, he provided an explicit construction of this hull, and he noted that, at least for ¯nite spaces, it comes endowed with an intrinsic polytopal cell structure. In this paper, we discuss Isbell's construction, we summarize the his- tory of | and some basic questions studied in | 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 strati¯cation of the cone of all metrics de- ¯ned on a ¯nite set X that is based on the combinatorial properties of the polytopal cell structure of Isbell's injective hulls and, on the other, various strati¯cations of the cone of positive semi-de¯nite quadratic forms de¯ned on Rn that were introduced by the Russian school in the context of reduction theory. 2000 Mathematics Subject Classi¯cation: 15A63, 05C05, 92-02, 92B99 1The author thanks the Swedish National Research Council (VR) for its support (grant# B 5107-20005097/2000). 2The author thanks the Swedish Natural Science Research Council (VR) for its support (grant# M12342-300) Documenta Mathematica ¢ Quadratic Forms LSU 2001 ¢ 121{139 122 A. Dress, K. T. Huber, V. Moulton Keywords and Phrases: injective hull, tight span, phylogenetic tree, quadratic form 1 Introduction The close relationship between distance analysis and quadratic-form theory was known already in pre-Pythagorean times: A ceramic slab found in the near east, for instance, presents the triples of integers 3,4,5; 5,12,13; 7,24,25; ... and it is very likely that these integers were of interest to Babylonean builders as they allowed to build walls at right angles without any particular tool except a long string with 12 = 3+4+5 or 30 = 5+12+13 or ... equidistant nodes (see Figure 1). ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ ¡ ¡ ¡ ¢¡¢¡¢¡¢ Figure 1: The ¯gure shows how a wall can be built at a right angle with a string of length 12. The Pythagorean Theorem puts this knowledge into more formal terms. And since then, the analysis of distance relationships has always been closely intertwined with that of quadratic forms. The development of di®erential geometry since Gauss as well as the development of geometric algebra in the 19th century | culminating in the de¯nition of Cli®ord and Cayley algebras and Hamilton's de¯nition of quarternian ¯elds | clearly testi¯es to this fact. In the early 20th century, attempts to develop appropriate conceptual frame- works for dealing with topological phenomena led Frechet to the de¯nition of metric spaces. While this caused most mathematicians to think of metric Documenta Mathematica ¢ Quadratic Forms LSU 2001 ¢ 121{139 Metric Spaces in Pure and Applied Mathematics 123 spaces as just a rather convenient tool to de¯ne and to deal with topological spaces, a few began to study metric spaces for their own sake. Menger and Blumenthal in particular began developing distance geometry providing and investigating necessary and su±cient conditions for a given metric space to be isometrically embeddable into standard metric spaces, e.g. the n-dimensional Euclidean or some hyperbolic, elliptic, or Lp space | rediscovering, by the way, an important result of Cayley's regarding the signi¯cance of the now famous Cayley-Menger determinants in this context3. While, at their time, this e®ort did not stimulate much of a response among their fellow mathematicians, it turned out to be crucial later on for developing algorithms that would identify the spatial structure of proteins from two-dimensional NMR data (cf. [7]). 2 John Isbell's Contribution Perhaps the ¯rst paper that studied metric spaces as such { without trying to study their embeddability into standard metric spaces nor looking at them as mere `presentations' of the underlying topological space { was, to our knowl- edge, written in the late sixties by John Isbell (cf. [23]). Trying to capture the decisive aspects of distance relationships, he proposed to de¯ne the category of metric spaces as follows: Its objects | for sure | are the metric spaces. But, noting that ² using continuous maps as morphisms would create too flexible a category, overemphasizing the topological aspects and neglecting the true metric structure (e.g. any bijection between two ¯nite metric spaces would then be an isomorphism) while ² using isometries only would result in too rigid a category without enough morphisms, he proposed to use the non-expansive maps from a metric space A into a metric space B as the set of morphisms from A to B, that is, those maps f : A ! B for which the distance in B between the image f(a) and f(a0) of two points a and a0 from A never exceeds their distance in A (or, in other words, the continuous maps from A to B for which the Weierstrass ± can always be chosen to be equal to the Weierstrass "). Isbell then went on to show that a unique injective hull exists in this category for every one of its objects, providing an explicit construction of this hull for all spaces and noting that it comes endowed, at least for ¯nite spaces, with an 3Actually, Cayley's original paper dealing with these determinants was the ¯rst to intro- duce the present notation for determinants. Documenta Mathematica ¢ Quadratic Forms LSU 2001 ¢ 121{139 124 A. Dress, K. T. Huber, V. Moulton intrinsic polytopal structure. More precisely, Isbell presented the following intriguing observations: (i) There exist injective metric spaces, that is, metric spaces X = (X; d) = 0 (X; d : X £X ! R) such that, for every isometric embedding ® : X ,! X of (X; d) into another metric space (X 0; d0), there exists a non-expansive retract ®0 : X0 ! X, that is, a non-expansive map ®0 from X0 into X 0 with ® ± ® = IdX : (ii) Every metric space (X; d) can be embedded isometrically into an injective metric space (X^; d^). (iii) Given any such isometric embedding ® : X ,! X^ of a metric space (X; d) into an injective metric space (X^; d^), there exists a unique smallest injec- tive subspace (X¹; d¹) of (X^; d^) containing ®(X). This subspace depends { up to isometry { only on (X; d): ² The map ¹ X ¹ X ! R : x¹ 7! (hx¹ : X ! R : x 7! d(®(x); x¹)) is easily seen to de¯ne an isometric embedding of X¹ into the set RX of all maps from X into R endowed with the supremum norm (or l1metric) X jjf; gjj1 := sup(jf(x) ¡ g(x)j : x 2 X) (f; g 2 R ): ² And its image consists exactly of all those maps f 2 RX that satisfy the condition f(x) = sup(d(x; y) ¡ f(y) : y 2 X) for all x 2 Xg. In [9], this subset of RX has also been called the tight span T (X; d) of (X; d) | a tradition that we will follow is in this paper, too. (iv) In addition, the above embedding identi¯es X with the set fhx : X ! R : y 7! d(y; x) : x 2 Xg and, hence, with the subset T 0(X; d) := ff 2 T (X; d) : 0 2 f(X)g of T (X; d). Documenta Mathematica ¢ Quadratic Forms LSU 2001 ¢ 121{139 Metric Spaces in Pure and Applied Mathematics 125 (v) The above de¯nition/construction of T (X; d) identi¯es it with a subset of the convex set X P (X; d) := ff 2 R : f(x) + f(y) ¸ d(x; y) for all x; y 2 Xg; more precisely, it identi¯es it with the set of all minimal maps in P (X; d) (relative to the partial order P (X; d) inherits from the partial order of RX de¯ned, as usual, by f · g () f(x) · g(x) for all x 2 X). Thus, it consists of a locally ¯nite collection of (low-dimensional) faces of P (X; d) whenever this convex set is actually a convex polytope (i.e. determined by a `locally ¯nite' collection of half spaces) which is surely the case if X itself is ¯nite. (v) T (X; d) is always contractible. More precisely, there exists always a con- tinuous family ft (t 2 [0; 1]) of non-expansive maps ft : T (X; d) ! T (X; d) with f0 = IdT (X;d) and #f1(T (X; d)) = 1. Although these notions may appear to be somewhat strange at ¯rst, the tight span of small metric spaces (X; d) can be described in simple geometric terms as follows: In case X consists of just two points of distance c, its tight span is exactly the interval of length c, its end points being just the two points from X (thus the name \tight span"). In case X consists of just three points of distance c1; c2; c3, its tight span is the union of three intervals of length (c1 +c2 ¡c3)=2; (c1 +c3 ¡c2)=2; and (c2 +c3 ¡c1)=2, respectively, all identi¯ed at one end point while the other three end points are the three points from X.
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