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Publication Details
AFRICAN RESEARCH NEXUS
SHINING A SPOTLIGHT ON AFRICAN RESEARCH
mathematics
Roundness properties of ultrametric spaces
Glasgow Mathematical Journal, Volume 56, No. 3, Year 2014
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Description
Motivated by a classical theorem of Schoenberg, we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space ℝn of dimension n but it cannot be isometrically embedded in any Euclidean space ℝr of dimension r < n. We use this result as a technical tool to study 'roundness' properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X,d): (1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0 and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class ℳ of all finite metric spaces that may be isometrically embedded into ℓ2 as an affinely independent set. The results of this paper show that Shkarin's class ℳ consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞]. Copyright © Glasgow Mathematical Journal Trust 2013.
Authors & Co-Authors
Faver, Timothy E.
United States, Philadelphia
Drexel University
Kochalski, Katelynn D.
United States, Charlottesville
University of Virginia
Murugan, Mathav Kishore
United States, Ithaca
Cornell University
Verheggen, Heidi
United States, Ithaca
Cornell University
Wesson, Elizabeth N.
United States, Ithaca
Cornell University
Weston, Anthony
United States, Buffalo
Canisius College
South Africa, Pretoria
University of South Africa
Statistics
Citations: 13
Authors: 6
Affiliations: 5
Identifiers
Doi:
10.1017/S0017089513000438
ISSN:
00170895
e-ISSN:
1469509X