Quasisymmetric Embeddings of Metric Spaces in Euclidean Space

2001 ◽  
pp. 98-102
Author(s):  
Juha Heinonen
Keyword(s):  
Euclidean Space ◽  
Metric Spaces

scholarly journals Simultaneously vanishing higher derived limits

2021 ◽  
Vol 9 ◽  
Author(s):  
Jeffrey Bergfalk ◽  
Chris Lambie-Hanson
Keyword(s):  
Euclidean Space ◽  
Metric Spaces ◽  
Inverse System ◽  
Finite Support ◽  
Partial Functions ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Strong Homology ◽  
Finite Support Iteration ◽  
Weakly Compact Cardinal

Abstract In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for every $n>0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for all $n>0$ . We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to $\lim ^n\textbf {A}=0$ will hold for each $n>0$ . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions $\mathbb {N}^2\to \mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:\mathbb {N}\to \mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .

scholarly journals ROUNDNESS PROPERTIES OF ULTRAMETRIC SPACES

2013 ◽  
Vol 56 (3) ◽  
pp. 519-535 ◽  
Author(s):  
TIMOTHY FAVER ◽  
KATELYNN KOCHALSKI ◽  
MATHAV KISHORE MURUGAN ◽  
HEIDI VERHEGGEN ◽  
ELIZABETH WESSON ◽  
...  
Keyword(s):  
Euclidean Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Independent Set ◽  
Ultrametric Space ◽  
Ultrametric Spaces ◽  
Negative Type ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Generalized Roundness

AbstractMotivated 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 ≥ 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 $\mathcal{M}$ 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 $\mathcal{M}$ 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, ∞].

A Menger Redux: Embedding Metric Spaces Isometrically in Euclidean Space

2017 ◽  
Vol 124 (7) ◽  
pp. 621 ◽  
Author(s):  
John C. Bowers ◽  
Philip L. Bowers
Keyword(s):  
Euclidean Space ◽  
Metric Spaces

Local behavior of mappings of metric spaces with branching

2020 ◽  
Vol 17 (4) ◽  
pp. 574-593
Author(s):  
Serhii Skvortsov
Keyword(s):  
Euclidean Space ◽  
Metric Spaces ◽  
Local Behavior ◽  
Compact Closure ◽  
Fixed Domain ◽  
Riemannian Sphere ◽  
Main Inequality

The local behavior of mappings with the inverse Poletsky inequality between metric spaces is studied. The case where one of the spaces satisfies the condition of weak sphericalization, is similar to the Riemannian sphere (extended Euclidean space), and is locally linearly connected under a mapping is considered. It is proved that the equicontinuity of the corresponding families of mappings of two domains, one of which is a domain with a weakly flat boundary, and another one is a fixed domain with a compact closure, the corresponding weight in the main inequality being supposed to be integrable.

scholarly journals PTOLEMAIC SPACES AND CAT(0)

2009 ◽  
Vol 51 (2) ◽  
pp. 301-314 ◽  
Author(s):  
S. M. BUCKLEY ◽  
K. FALK ◽  
D. J. WRAITH
Keyword(s):  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Metric Space ◽  
Metric Spaces ◽  
Finsler Manifold ◽  
Full Generality ◽  
Space Setting ◽  
Ptolemaic Space

AbstractWe consider Ptolemy's inequality in a metric space setting. It is not hard to see that CAT(0) spaces satisfy this inequality. Although the converse is not true in full generality, we show that if our Ptolemaic space is either a Riemannian or Finsler manifold, then it must also be CAT(0). Ptolemy's inequality is closely related to inversions of metric spaces. We exploit this link to establish a new characterization of Euclidean space amongst all Riemannian manifolds.

scholarly journals Metric spaces which cannot be isometrically embedded in Hilbert space

1984 ◽  
Vol 30 (2) ◽  
pp. 161-167
Author(s):  
Yang Lu ◽  
Zhang Jing-Zhong
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Metric Spaces ◽  
Geometric Inequality ◽  
Cambridge Philos ◽  
Higher Dimensional ◽  
Planar Convex ◽  
Convex Quadrangle

Let A1A2A3A4, be a planar convex quadrangle with diagonals A1A3 and A2A4. Is there a quadrangle B1B2B3B4 in Euclidean space such that A1A3 < B1B3, A2A4 < B2B4 but AiAj > BiBj for other edges?The answer is “no”. It seems to be obvious but the proof is more difficult. In this paper we shall solve similar more complicated problems by using a higher dimensional geometric inequality which is a generalisation of the well-known Pedoe inequality (Proc. Cambridge Philos. Soc.38 (1942), 397–398) and an interesting result by L.M. Blumenthal and B.E. Gillam (Amer. Math. Monthly50 (1943), 181–185).

RECTIFIABILITY OF MEASURES WITH LOCALLY UNIFORM CUBE DENSITY

2003 ◽  
Vol 86 (1) ◽  
pp. 153-249 ◽  
Author(s):  
ANDREW LORENT
Keyword(s):  
Euclidean Space ◽  
Isometric Immersion ◽  
Metric Spaces ◽  
Model Problem ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Initial Condition ◽  
Radon Measures ◽  
Geometric Measure ◽  
Euclidean Unit Ball

The conjecture that Radon measures in Euclidean space with positive finite density are rectifiable was a central problem in Geometric Measure Theory for fifty years. This conjecture was positively resolved by Preiss in 1986, using methods entirely dependent on the symmetry of the Euclidean unit ball. Since then, due to reasons of isometric immersion of metric spaces into $l_{\infty}$ and the uncommon nature of the sup norm even in finite dimensions, a popular model problem for generalising this result to non-Euclidean spaces has been the study of 2-uniform measures in $l^{3}_{\infty}$. The rectifiability or otherwise of these measures has been a well-known question.In this paper the stronger result that locally 2-uniform measures in $l^{3}_{\infty}$ are rectifiable is proved. This is the first result that proves rectifiability, from an initial condition about densities, for general Radon measures of dimension greater than 1 outside Euclidean space.2000 Mathematical Subject Classification: 28A75.

scholarly journals On Ptolemaic metric simplicial complexes

2010 ◽  
Vol 149 (1) ◽  
pp. 93-104 ◽  
Author(s):  
S. M. BUCKLEY ◽  
D. J. WRAITH ◽  
J. McDOUGALL
Keyword(s):  
Euclidean Space ◽  
Simplicial Complex ◽  
Large Class ◽  
Metric Spaces ◽  
Simplicial Complexes ◽  
Mild Conditions

AbstractWe show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy's inequality is closely related to inversions of metric spaces. For a large class of metric simplicial complexes, we characterize those which are isometric to Euclidean space in terms of metric inversions.

Embedding metric spaces in Euclidean space

1974 ◽  
Vol 5 (1) ◽  
pp. 101-107 ◽  
Author(s):  
C. L. Morgan
Keyword(s):  
Euclidean Space ◽  
Metric Spaces
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