Finite metric spaces of strictly negative type

Hjorth, Poul G.; Lisonĕk, P; Markvorsen, Steen; Thomassen, Carsten

doi:10.1016/s0024-3795(97)00242-5

Cited by 39 publications

(34 citation statements)

References 13 publications

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“…It is known that all finite subspaces of Euclidean space R m are hypermetric and regular and hence of strictly negative type. This is also true for trees; see [4]. S m is hypermetric and hence of negative type.…”

Section: Definitionsmentioning

confidence: 84%

“…[5]. In [4] it was shown that regularity of the distance matrix together with hypermetricity implied strictly negative type. Thus, for hypermetric spaces, regularity is equivalent to being of strictly negative type.…”

Section: Definitionsmentioning

confidence: 99%

“…S m is hypermetric and hence of negative type. In [4] the subspaces of S m of strictly negative type are identified as those containing at most one pair of antipodal points.…”

Section: Definitionsmentioning

confidence: 99%

“…It is a property sufficient to ensure uniqueness of the so-called ∞-extender, a constellation of weighted points realizing the transfinite diameter of the finite metric space. For a discussion of these concepts see [4].…”

Section: Introductionmentioning

confidence: 99%

“…The spheres S m are not (a discussion of S m in this connection is given below). It has been conjectured ( [4]) that the hyperbolic spaces H m R are of strictly negative type. This conjecture is settled, in the affirmative, by Corollary 4.2 of section 4.…”

Section: Introductionmentioning

confidence: 99%

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Hyperbolic spaces are of strictly negative type

Hjorth¹,

Kokkendorff²,

Markvorsen³

2001

Proc. Amer. Math. Soc.

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Abstract. We study finite metric spaces with elements picked from, and distances consistent with, ambient Riemannian manifolds. The concepts of negative type and strictly negative type are reviewed, and the conjecture that hyperbolic spaces are of strictly negative type is settled, in the affirmative. The technique of the proof is subsequently applied to show that every compact manifold of negative type must have trivial fundamental group, and to obtain a necessary criterion for product manifolds to be of negative type.

show abstract

Section: Definitionsmentioning

confidence: 84%

Section: Definitionsmentioning

confidence: 99%

“…S m is hypermetric and hence of negative type. In [4] the subspaces of S m of strictly negative type are identified as those containing at most one pair of antipodal points.…”

Section: Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Hyperbolic spaces are of strictly negative type

Hjorth¹,

Kokkendorff²,

Markvorsen³

2001

Proc. Amer. Math. Soc.

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The Brownian Castle

Cannizzaro

Hairer

2022

Comm Pure Appl Math

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We introduce a 1 g 1-dimensional temperature-dependent model such that the classical ballistic deposition model is recovered as its zero-temperature limit. Its I-temperature version, which we refer to as the 0-Ballistic Deposition (0-BD) model, is a randomly evolving interface which, surprisingly enough, does not belong to either the Edwards-Wilkinson (EW) or the Kardar-Parisi-Zhang (KPZ) universality class. We show that 0-BD has a scaling limit, a new stochastic process that we call Brownian Castle (BC) which, although it is "free", is distinct from EW and, like any other renormalisation fixed point, is scale-invariant, in this case under the 1 1 2 scaling (as opposed to 1 2 3 for KPZ and 1 2 4 for EW). In the present article, we not only derive its finite-dimensional distributions, but also provide a "global" construction of the Brownian Castle which has the advantage of highlighting the fact that it admits backward characteristics given by the (backward) Brownian Web (see [16,37]). Among others, this characterisation enables us to establish fine pathwise properties of BC and to relate these to special points of the Web. We prove that the Brownian Castle is a (strong) Markov and Feller process on a suitable space of càdlàg functions and determine its long-time behaviour. Finally, we give a glimpse to its universality by proving the convergence of 0-BD to BC in a rather strong sense.

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Distance geometry in quasihypermetric spaces. II

Nickolas

Wolf

2010

Mathematische Nachrichten

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Abstract. Let (X, d) be a compact metric space and let M(X) denote the space of all finite signed Borel measures on X. Define I : M(X) → R byand set M (X) = sup I(µ), where µ ranges over the collection of signed measures in M(X) of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III ], investigates the geometric constant M (X) and its relationship to the metric properties of X and the functional-analytic properties of a certain subspace of M(X) when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum defining M (X), sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the finiteness of M (X).

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Finite metric spaces of strictly negative type

Cited by 39 publications

References 13 publications

Hyperbolic spaces are of strictly negative type

Hyperbolic spaces are of strictly negative type

The Brownian Castle

Distance geometry in quasihypermetric spaces. II

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