By Michel Marie Deza

This up to date and revised moment variation of the major reference quantity on distance metrics incorporates a wealth of latest fabric that displays advances in a box now considered as an important device in lots of parts of natural and utilized mathematics.

The e-book of this quantity coincides with intensifying learn efforts into metric areas and particularly distance layout for functions. exact metrics became a vital aim in computational biology, photo research, speech acceptance and data retrieval.

Leaving apart the sensible questions that come up through the choice of a ‘good’ distance functionality, this paintings makes a speciality of delivering the learn neighborhood with a useful accomplished directory of the most on hand distances.

As good as delivering standalone introductions and definitions, the encyclopedia enables quick cross-referencing with simply navigable bold-faced textual hyperlinks to center entries. as well as distances themselves, the authors have collated various attention-grabbing curiosities of their Who’s Who of metrics, together with distance-related notions and paradigms that permit utilized mathematicians in different sectors to set up learn instruments that non-specialists justly view as arcane. In increasing entry to those concepts, and in lots of circumstances enriching the context of distances themselves, this peerless quantity is sure to stimulate clean research.

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This up-to-date and revised moment variation of the top reference quantity on distance metrics contains a wealth of recent fabric that displays advances in a box now considered as a necessary software in lots of parts of natural and utilized arithmetic. The book of this quantity coincides with intensifying study efforts into metric areas and particularly distance layout for purposes.

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The Steiner ratio St(X, d) of the metric space (X, d) is defined by d(SMT V ) . 4 Metric Mappings 33 For any metric space (X, d) we have R2 , 1 2 ≤ St (X, d) ≤ 1. For the l2 - and l1 -metric √ it is equal to 23 and 23 , respectively. (cf. Lp -metric in Chap. 5) on Cf. arc routing problems in Chap. 15. , the graph with the vertex-set X and the edge-set {xy : d(x, y) ∈ D}. Usually, (X, d) is an lp -space and D = {1} (Benda–Perles chromatic number) or D = [1 − , 1 + ] (the chromatic number of the -unit distance graph).

6. Metric fibration Given a complete metric space (X, d), two subsets M1 and M2 of X are called equidistant if for each x ∈ M1 there exists y ∈ M2 with d(x, y) being equal to the Hausdorff metric between the sets M1 and M2 . A metric fibration of (X, d) is a partition F of X into isometric mutually equidistant closed sets. The quotient metric space X/F inherits a natural metric for which the distance map is a submetry. , a bijective function f : X → Y such that f and f −1 are continuous (the preimage of every open set in Y is open in X).

The number lg2 P is called the metric capacity (or -capacity) of (X, d). • Steiner ratio Given a metric space (X, d) and a finite subset V ⊂ X, let G = (V , E) be the complete weighted graph on V with edge-weights d(x, y) for all x, y ∈ V . A spanning tree T in G is a subset of |V | − 1 edges forming a tree on V with the weight d(T ) equal to the sum of the weights of its edges. , a spanning tree in G with the minimal weight d(MST V ). A minimum Steiner tree on V is a tree SMT V such that its vertex-set is a subset of X containing V , and d(SMT V ) = infM⊂X:V ⊂M d(MST M ).