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Corrigendum to “Enhanced negative type for finite metric trees
 J. Funct. Anal
"... A finite metric tree is a finite connected graph that has no cycles, endowed with an edge weighted path metric. Finite metric trees are known to have strict 1negative type. In this paper we introduce a new family of inequalities (1) that encode the best possible quantification of the strictness of ..."
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A finite metric tree is a finite connected graph that has no cycles, endowed with an edge weighted path metric. Finite metric trees are known to have strict 1negative type. In this paper we introduce a new family of inequalities (1) that encode the best possible quantification of the strictness of the non trivial 1negative type inequalities for finite metric trees. These inequalities are sufficiently strong to imply that any given finite metric tree (T,d) must have strict pnegative type for all p in an open interval (1 − ζ,1 + ζ), where ζ> 0 may be chosen so as to depend only upon the unordered distribution of edge weights that determine the path metric d on T. In particular, if the edges of the tree are not weighted, then it follows that ζ depends only upon the number of vertices in the tree. We also give an example of an infinite metric tree that has strict 1negative type but does not have pnegative type for any p> 1. This shows that the maximal pnegative type of a metric space can be strict. Key words: Finite metric trees, strict negative type, generalized roundness
Finite dimensional subspaces of L_p
"... this article, we chose to devote this section to describing the change of densities that arise later. It turns out that the framework in which this technique is most naturally used is that of an L p () space when is a probability. For us there is no loss of generality in restricting to that case si ..."
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Cited by 10 (2 self)
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this article, we chose to devote this section to describing the change of densities that arise later. It turns out that the framework in which this technique is most naturally used is that of an L p () space when is a probability. For us there is no loss of generality in restricting to that case since the space #
Manifestations of non linear roundness in analysis, discrete geometry and topology
 in Limits of Graphs in Group Theory and Computer Science (Editors G. Arzhantseva and A. Valette), Research Proceedings of the École Polytechnique Fédérale de Lausanne
, 2009
"... Some forty years ago Per Enflo introduced the nonlinear notions of roundness and generalized roundness for general metric spaces in order to study (a) uniform homeomorphisms between (quasi) Banach spaces, and (b) Hilbert’s Fifth Problem in the context of non locally compact topological groups (see ..."
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Cited by 9 (6 self)
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Some forty years ago Per Enflo introduced the nonlinear notions of roundness and generalized roundness for general metric spaces in order to study (a) uniform homeomorphisms between (quasi) Banach spaces, and (b) Hilbert’s Fifth Problem in the context of non locally compact topological groups (see [24], [25], [26], and [27]). Since then the concepts of roundness and generalized roundness have proven to be particularly useful and durable across a number of important mathematical fields such as coarse geometry, discrete geometry, functional analysis and topology. The purpose of this article is to take a retrospective look at some notable applications of versions of nonlinear roundness across such fields, to draw some hitherto unpublished connections between such results, and to highlight some very intriguing open problems. 1. Nonlinear Roundness. Introduction and Background Nonlinear notions of roundness and generalized roundness (Definition 1.1) were introduced by Enflo in the late 1960s in a series of concise but elegant papers [24], [25], [26] and [27]. The purpose of Enflo’s program of study in these papers
Similarity, kernels, and the triangle inequality
 OURNAL OF MATHEMATICAL PSYCHOLOGY 52(5) 297303 (2008)
, 2008
"... Similarity is used as an explanatory construct throughout psychology and multidimensional scaling (MDS) is the most popular way to assess similarity. In MDS similarity is intimately connected to the idea of a geometric representation of stimuli in a perceptual space. Whilst connecting similarity and ..."
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Cited by 8 (1 self)
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Similarity is used as an explanatory construct throughout psychology and multidimensional scaling (MDS) is the most popular way to assess similarity. In MDS similarity is intimately connected to the idea of a geometric representation of stimuli in a perceptual space. Whilst connecting similarity and closeness of stimuli in a geometric representation may be intuitively plausible, Tversky and Gati (1982) have reported data which are inconsistent with the usual geometric representations that are based on segmental additivity. We show that similarity measures based on Shepard’s universal law of generalization (Shepard, 1987) lead to an inner product representation in a reproducing kernel Hilbert space. In such a space stimuli are represented by their similarity to all other stimuli. This representation, based on Shepard’s law, has a natural metric that does not have additive segments whilst still retaining the intuitive notion of connecting similarity and distance between stimuli. Furthermore, this representation has the psychologically appealing property that the distance between stimuli is bounded.
BANACH SPACES EMBEDDING ISOMETRICALLY INTO Lp WHEN 0 < p < 1
"... Abstract. For 0 < p < 1 we give examples of Banach spaces isometrically embedding into Lp but not into any Lr with p < r ≤ 1. 1. ..."
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Abstract. For 0 < p < 1 we give examples of Banach spaces isometrically embedding into Lp but not into any Lr with p < r ≤ 1. 1.
SMALL SUBSPACES OF Lp
"... Abstract. We prove that if X is a subspace of Lp (2 < p < ∞), then either X embeds isomorphically into ℓp ⊕ ℓ2 or X contains a subspace Y, which is isomorphic to ℓp(ℓ2). We also give an intrinsic characterization of when X embeds into ℓp ⊕ℓ2 in terms of weakly null trees in X or, equivalently, ..."
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Cited by 5 (0 self)
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Abstract. We prove that if X is a subspace of Lp (2 < p < ∞), then either X embeds isomorphically into ℓp ⊕ ℓ2 or X contains a subspace Y, which is isomorphic to ℓp(ℓ2). We also give an intrinsic characterization of when X embeds into ℓp ⊕ℓ2 in terms of weakly null trees in X or, equivalently, in terms of the “infinite asymptotic game ” played in X. This solves problems concerning small subspaces of Lp originating in the 1970’s. The techniques used were developed over several decades, the most recent being that of weakly null trees developed in the 2000’s. 1.
Cycles and 1unconditional matrices
, 2008
"... We characterize the 1unconditional subsets (erc)(r,c)∈I of the set of elementary matrices in the SchattenvonNeumann class S p. The set of couples I must be the set of edges of a bipartite graph without cycles of even length 4 � l � p if p is an even integer, and without cycles at all if p is a po ..."
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Cited by 5 (1 self)
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We characterize the 1unconditional subsets (erc)(r,c)∈I of the set of elementary matrices in the SchattenvonNeumann class S p. The set of couples I must be the set of edges of a bipartite graph without cycles of even length 4 � l � p if p is an even integer, and without cycles at all if p is a positive real number that is not an even integer. In the latter case, I is even a Varopoulos set of Vinterpolation of constant 1. We also study the metric unconditional approximation property for the space S p I spanned by (erc)(r,c)∈I in S p. 1
Sobolev spaces with only trivial isometries
 Positivity
"... Abstract. We will give some conditions for Sobolev spaces on bounded Lipschitz domains to admit only trivial isometries. 1. ..."
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Abstract. We will give some conditions for Sobolev spaces on bounded Lipschitz domains to admit only trivial isometries. 1.
LIPSCHITZ EXTENSION CONSTANTS EQUAL PROJECTION CONSTANTS
, 2006
"... Abstract. For a Banach space V we define its Lipschitz extension constant, LE(V), to be the infimum of the constants c such that for every metric space (Z, ρ), every X ⊂ Z, and every f: X → V, there is an extension, g, of f to Z such that L(g) ≤ cL(f), where L denotes the Lipschitz constant. The ba ..."
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Abstract. For a Banach space V we define its Lipschitz extension constant, LE(V), to be the infimum of the constants c such that for every metric space (Z, ρ), every X ⊂ Z, and every f: X → V, there is an extension, g, of f to Z such that L(g) ≤ cL(f), where L denotes the Lipschitz constant. The basic theorem is that when V is finitedimensional we have LE(V) = PC(V) where PC(V) is the wellknown projection constant of V. We obtain some direct consequences of this theorem, especially when V = Mn(C). We then apply known techniques for calculating projection constants, involving averaging of projections, to calculate LE((Mn(C)) sa). We also discuss what happens if we also require that ‖g‖ ∞ = ‖f‖∞. In my exploration of the relationship between vector bundles and Gromov– Hausdorff distance [20] I need to be able to extend matrixvalued functions from a closed subset of a compact metric space to the whole metric space, with as little increase of the Lipschitz constant as possible. There is a substantial literature concerned with extending Lipschitz functions, but I have had difficulty finding there