On the Generalized Roundness of Finite Metric Spaces

Weston, Anthony

doi:10.1006/jmaa.1995.1174

Cited by 13 publications

(14 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As an immediate consequence of Corollary 4.3, Theorem 5.4, [30,Theorem 4.3] and the examples given in [11,Section 1] we obtain the following theorem. By way of marked contrast with Theorem 5.5, we note that (a) the set of all values of p for which the Banach space C[0, 1] has strict p-negative type is the degenerate interval {0} (this follows from [11, Theorem 2.1] and Theorem 2.5), and (b) the supremal p-negative type of an infinite metric space may or may not be strict.…”

Section: Range Of Strict P-negative Typementioning

confidence: 60%

See 1 more Smart Citation

Strict p-negative type of a metric space

Weston

2009

Positivity

View full text Add to dashboard Cite

Abstract. Doust and Weston [8] have introduced a new method called enhanced negative type for calculating a non-trivial lower bound ℘ T on the supremal strict p-negative type of any given finite metric tree (T, d). In the context of finite metric trees any such lower bound ℘ T > 1 is deemed to be non-trivial. In this paper we refine the technique of enhanced negative type and show how it may be applied more generally to any finite metric space (X, d) that is known to have strict p-negative type for some p ≥ 0. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given in [8, Corollary 5.5] and, moreover, leads in to one of our main results: The supremal p-negative type of a finite metric space cannot be strict. By way of application we are then able to exhibit large classes of finite metric spaces (such as finite isometric subspaces of Hadamard manifolds) that must have strict p-negative type for some p > 1. We also show that if a metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q ∈ [0, p). This generalizes Schoenberg [27, Theorem 2] and leads to a complete classification of the intervals on which a metric space may have strict p-negative type.

show abstract

Section: Range Of Strict P-negative Typementioning

confidence: 60%

“…Rather than giving the original definition of generalized roundness p from [11], we shall present an equivalent reformulation in Definition 2.4 (a) that is due to Lennard et al [20] and Weston [30].…”

Section: A Framework For Ordinary and Strict P-negative Typementioning

confidence: 99%

Strict p-negative type of a metric space

Weston

2009

Positivity

View full text Add to dashboard Cite

show abstract

“…Theorem 2.1 is a variation of themes developed in Weston [16], Lennard et al [9], and Doust and Weston [4]. The motivation for all such studies has stemmed from Enflo's original definition of generalized roundness that was given in [5].…”

Section: Signed Simplices and Polygonal Equalities In L P -Spacesmentioning

confidence: 97%

The geometry of two-valued subsets of L_p -spaces

Weston

2017

Mathematica Slovaca

View full text Add to dashboard Cite

Let M(Ω, µ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, µ). Let B ⊂ M(Ω, µ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0, 1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p-negative type when viewed as a metric subspace of Lp(Ω, µ) if and only if B is an affinely independent subset of M(Ω, µ) (when M(Ω, µ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of Lp(Ω, µ) has strict p-negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in Lp[0, 1] is seen to have strict p-negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of Lp(Ω, µ) that have p-negative type but not q-negative type for any q > p. Such sets preclude the existence of certain types of isometry into Lp-spaces.

show abstract

“…Simplices with weights on the vertices were introduced by Weston [22] to study the generalized roundness of finite metric spaces. In [22] the author only considers positive weights. The approach being taken here appears to be more general but it is, in fact, equivalent by Lemma 3.13.…”

Section: Polygonal Equalities In Metric Spacesmentioning

confidence: 99%

Polygonal equalities and virtual degeneracy inLp-spaces

Kelleher

Miller

Osborn

et al. 2014

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Suppose 0 < p ≤ 2 and that (Ω, µ) is a measure space for which Lp(Ω, µ) is at least twodimensional. The central results of this paper provide a complete description of the subsets of Lp(Ω, µ) that have strict p-negative type. In order to do this we study non-trivial p-polygonal equalities in Lp(Ω, µ). These are equalities that can, after appropriate rearrangement and simplification, be expressed in the formwhere {z 1 , . . . , zn} is a subset of Lp(Ω, µ) and α 1 , . . . , αn are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial p-polygonal equalities in Lp(Ω, µ). The cases p < 2 and p = 2 are substantially different and are treated separately. The case p = 1 generalizes an elegant result of Elsner, Han, Koltracht, Neumann and Zippin.Another reason for studying non-trivial p-polygonal equalities in Lp(Ω, µ) is due to the fact that they preclude the existence of certain types of isometry. For example, our techniques show that if (X, d) is a metric space that has strict q-negative type for some q ≥ p, then: (1) (X, d) is not isometric to any linear subspace W of Lp(Ω, µ) that contains a pair of disjointly supported non-zero vectors, and (2) (X, d) is not isometric to any subset of Lp(Ω, µ) that has non-empty interior. Furthermore, in the case p = 2, it also follows that (X, d) is not isometric to any affinely dependent subset of L 2 (Ω, µ). More generally, we show that if (Y, ρ) is a metric space whose generalized roundness ℘ is finite and if (X, d) is a metric space that has strict q-negative type for some q ≥ ℘, then (X, d) is not isometric to any metric subspace of (Y, ρ) that admits a non-trivial p 1 -polygonal equality for some p 1 ∈ [℘, q]. It is notable in all of these statements that the metric space (X, d) can, for instance, be any ultrametric space. As a result we obtain new insights into sophisticated embedding theorems of Lemin and Shkarin.We conclude the paper by constructing some pathological infinite-dimensional linear subspaces of ℓp that do not have strict p-negative type.2010 Mathematics Subject Classification. 54E40, 46B04, 46C05.

show abstract

On the Generalized Roundness of Finite Metric Spaces

Cited by 13 publications

References 0 publications

Strict p-negative type of a metric space

Strict p-negative type of a metric space

The geometry of two-valued subsets of L_p -spaces

Polygonal equalities and virtual degeneracy inLp-spaces

Contact Info

Product

Resources

About

On the Generalized Roundness of Finite Metric Spaces

Cited by 13 publications

References 0 publications

Strict p-negative type of a metric space

Strict p-negative type of a metric space

The geometry of two-valued subsets of Lp -spaces

Polygonal equalities and virtual degeneracy inLp-spaces

Contact Info

Product

Resources

About

The geometry of two-valued subsets of L_p -spaces