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Euclidean Quotients of Finite Metric Spaces
"... This paper is devoted to the study of quotients of finite metric spaces. The basic typeof question we ask is: Given a finite metric space M and ff> = 1, what is the largest quotientof (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tightbounds for these questions, ..."
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Cited by 40 (19 self)
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This paper is devoted to the study of quotients of finite metric spaces. The basic typeof question we ask is: Given a finite metric space M and ff> = 1, what is the largest quotientof (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tightbounds for these questions
Finite Metrics in Switching Classes
"... Let D be a finite set and g: D × D → R a symmetric function satisfying g(x, x) = 0 and g(x, y) = g(y, x) for all x, y ∈ D. A switch g σ is obtained from g by using a local valuation σ: D → R: g σ (x, y) = σ(x) + g(x, y) + σ(y) for x ̸ = y. It is shown that every symmetric function g has a unique ..."
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minimal pseudometric switch, and, moreover, there is a switch g σ of g that is isometric to a finite Manhattan metric. Also, for each metric on a finite set D, we associate an extension metric on the set of all nonempty subsets of D, and we show that this extended metric inherits the switching classes
Equilateral Triangles in Finite Metric Spaces
, 2004
"... In the context of finite metric spaces with integer distances, we investigate the new Ramsey-type question of how many points can a space contain and yet be free of equilateral triangles. In particular, for finite metric spaces with distances in the set {1,...,n},thenumberD n is defined as the ..."
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Cited by 1 (0 self)
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In the context of finite metric spaces with integer distances, we investigate the new Ramsey-type question of how many points can a space contain and yet be free of equilateral triangles. In particular, for finite metric spaces with distances in the set {1,...,n},thenumberD n is defined
25.1 Finite Metric Spaces
"... d(x, y) + d(y, z) ≥ d(x, z) (triangle inequality). For example, IR 2 with the regular Euclidean distance is a metric space. It is usually of interest to consider the finite case, where X is an a set of n points. Then, the function d can be specified by � � n real numbers; that is, the distance betw ..."
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d(x, y) + d(y, z) ≥ d(x, z) (triangle inequality). For example, IR 2 with the regular Euclidean distance is a metric space. It is usually of interest to consider the finite case, where X is an a set of n points. Then, the function d can be specified by � � n real numbers; that is, the distance
Finite Metric Spaces - Combinatorics, Geometry and Algorithms
- In Proceedings of the International Congress of Mathematicians III
, 2002
"... This article deals only with what might be called the geometrization of combinatorics. Namely, the idea that viewing combinatorial objects from a geometric perspective often yields unexpected insights. Even more concretely, we concentrate on finite metric spaces and their embeddings ..."
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Cited by 49 (2 self)
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This article deals only with what might be called the geometrization of combinatorics. Namely, the idea that viewing combinatorial objects from a geometric perspective often yields unexpected insights. Even more concretely, we concentrate on finite metric spaces and their embeddings
Embeddings of locally finite metric spaces into Banach spaces
- Proc. Amer. Math. Soc
"... (Communicated by N. Tomczak-Jaegermann) Abstract. We show that if X is a Banach space without cotype, then every locally finite metric space embeds metrically into X. 1. ..."
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Cited by 14 (7 self)
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(Communicated by N. Tomczak-Jaegermann) Abstract. We show that if X is a Banach space without cotype, then every locally finite metric space embeds metrically into X. 1.
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