Strict p-negative type of a metric space

Strict p-negative type of a metric space Doust and Weston (J Funct Anal 254:2336–2364, 2008) have introduced a new method called enhanced negative type for calculating a non-trivial lower bound $${\wp_{T}}$$ 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 $${\wp_{T} >1 }$$ 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 Doust and Weston (J Funct Anal 254:2336–2364, 2008, 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 $${q \in [0, p)}$$ . This generalizes Schoenberg (Ann Math 38:787–793, 1937, Theorem 2) and leads to a complete classification of the intervals on which a metric space may have strict p-negative type. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Strict p-negative type of a metric space

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Publisher
Springer Journals
Copyright
Copyright © 2009 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-009-0035-2
Publisher site
See Article on Publisher Site

Abstract

Doust and Weston (J Funct Anal 254:2336–2364, 2008) have introduced a new method called enhanced negative type for calculating a non-trivial lower bound $${\wp_{T}}$$ 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 $${\wp_{T} >1 }$$ 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 Doust and Weston (J Funct Anal 254:2336–2364, 2008, 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 $${q \in [0, p)}$$ . This generalizes Schoenberg (Ann Math 38:787–793, 1937, Theorem 2) and leads to a complete classification of the intervals on which a metric space may have strict p-negative type.

Journal

PositivitySpringer Journals

Published: Nov 17, 2009

References

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