Roundness properties of ultrametric spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1311.46020
DOI: 10.1017/S0017089513000438
author: Timothy Faver
author: Katelynn Kochalski
author: Mathav Kishore Murugan
author: Heidi Verheggen
author: Elizabeth Wesson
author: Anthony Weston
published in: Glasgow Mathematical Journal
publication date: 2014-09-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.6669
Mathematics Subject Classification ID: 46B85
Mathematics Subject Classification ID: 05C12
Mathematics Subject Classification ID: 30L05
Mathematics Subject Classification ID: 46C05
Mathematics Subject Classification ID: 51K05
Mathematics Subject Classification ID: 54E40
zbMATH DE Number: 6349684
zbMATH Keywords: isometry
zbMATH Keywords: strict negative type
zbMATH Keywords: generalized roundness
zbMATH Keywords: additive metric
zbMATH Keywords: leaf metric
zbMATH Keywords: ultrametric
MaRDI profile type: Publication
arXiv ID: 1201.6669
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OpenAlex ID: W2089901684
title: Roundness properties of ultrametric spaces
review text: The paper is devoted to the geometry of metric spaces of some special types. The main classes of considered metrics are: ultrametrics, tree metrics (also known as additive metrics) and leaf metrics (leaf metrics are defined as restrictions of metrics of metric trees to their sets of leaves).NEWLINENEWLINEThe geometric properties studied in the paper are (in a sense) centered around the notion of \(p\)-negative type defined as follows: A metric space \((X,d)\) has \(p\)-negative type if and only if for all finite subsets \(\{x_{1}, \dots , x_{n} \} \subseteq X\) and all choices of real numbers \(\eta_{1}, \dots, \eta_{n}\) with \(\eta_{1} + \cdots + \eta_{n} = 0\), the condition \(\sum_{1 \leq i,j \leq n} d(x_{i},x_{j})^{p} \eta_{i} \eta_{j} \leq 0\) holds. The paper contains numerous results on geometric properties of the listed classes of metric spaces.NEWLINENEWLINE The authors' abstract describes some of the most important of them: ``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 \(\mathbb{R}^{n}\) of dimension \(n\) but it cannot be isometrically embedded in any Euclidean space \(\mathbb{R}^{r}\) 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 \(p \geq 0\), and (5) \(X\) admits no \(p\)-polygonal equality for any \(p \geq 0\). 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 se…
reviewed by: Mikhail Ostrovskii
description: scientific article; zbMATH DE number 6349684