Ramsey-like properties for bi-Lipschitz mappings of finite metric spaces

Jiří Matoušek

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 3, page 451-463
  • ISSN: 0010-2628

Abstract

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Let ( X , ρ ) , ( Y , σ ) be metric spaces and f : X → Y an injective mapping. We put ∥ f ∥ L i p = sup { σ ( f ( x ) , f ( y ) ) / ρ ( x , y ) ; x , y ∈ X , x ≠ y } , and dist ( f ) = ∥ f ∥ L i p . ∥ f - 1 ∥ L i p (the distortion of the mapping f ). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let X be a finite metric space, and let ε > 0 , K be given numbers. Then there exists a finite metric space Y , such that for every mapping f : Y → Z ( Z arbitrary metric space) with dist ( f ) < K one can find a mapping g : X → Y , such that both the mappings g and f | g ( X ) have distortion at most ( 1 + ε ) . If X is isometrically embeddable into a ℓ p space (for some p ∈ [ 1 , ∞ ] ), then also Y can be chosen with this property.

How to cite

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Matoušek, Jiří. "Ramsey-like properties for bi-Lipschitz mappings of finite metric spaces." Commentationes Mathematicae Universitatis Carolinae 33.3 (1992): 451-463. <http://eudml.org/doc/247391>.

@article{Matoušek1992,
abstract = {Let $(X,\rho )$, $(Y,\sigma )$ be metric spaces and $f:X\rightarrow Y$ an injective mapping. We put $\Vert f\Vert _\{Lip\} = \sup \lbrace \sigma (f(x),f(y))/\rho (x,y)$; $x,y\in X$, $x\ne y\rbrace $, and $\operatorname\{dist\}(f)= \Vert f\Vert _\{Lip\}.\Vert f^\{-1\}\Vert _\{Lip\}$ (the distortion of the mapping $f$). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let $X$ be a finite metric space, and let $\varepsilon >0$, $K$ be given numbers. Then there exists a finite metric space $Y$, such that for every mapping $f:Y\rightarrow Z$ ($Z$ arbitrary metric space) with $\operatorname\{dist\}(f)<K$ one can find a mapping $g:X\rightarrow Y$, such that both the mappings $g$ and $f|_\{g(X)\}$ have distortion at most $(1+\varepsilon )$. If $X$ is isometrically embeddable into a $\ell _p$ space (for some $p\in [1,\infty ]$), then also $Y$ can be chosen with this property.},
author = {Matoušek, Jiří},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Ramsey theory; embedding of metric spaces; distortion; Lipschitz mapping; differentiability of Lipschitz mappings; Ramsey theory; embedding of metric spaces; Lipschitz mapping; differentiability of Lipschitz mappings; distortion; finite metric space},
language = {eng},
number = {3},
pages = {451-463},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Ramsey-like properties for bi-Lipschitz mappings of finite metric spaces},
url = {http://eudml.org/doc/247391},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Matoušek, Jiří
TI - Ramsey-like properties for bi-Lipschitz mappings of finite metric spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 3
SP - 451
EP - 463
AB - Let $(X,\rho )$, $(Y,\sigma )$ be metric spaces and $f:X\rightarrow Y$ an injective mapping. We put $\Vert f\Vert _{Lip} = \sup \lbrace \sigma (f(x),f(y))/\rho (x,y)$; $x,y\in X$, $x\ne y\rbrace $, and $\operatorname{dist}(f)= \Vert f\Vert _{Lip}.\Vert f^{-1}\Vert _{Lip}$ (the distortion of the mapping $f$). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let $X$ be a finite metric space, and let $\varepsilon >0$, $K$ be given numbers. Then there exists a finite metric space $Y$, such that for every mapping $f:Y\rightarrow Z$ ($Z$ arbitrary metric space) with $\operatorname{dist}(f)<K$ one can find a mapping $g:X\rightarrow Y$, such that both the mappings $g$ and $f|_{g(X)}$ have distortion at most $(1+\varepsilon )$. If $X$ is isometrically embeddable into a $\ell _p$ space (for some $p\in [1,\infty ]$), then also $Y$ can be chosen with this property.
LA - eng
KW - Ramsey theory; embedding of metric spaces; distortion; Lipschitz mapping; differentiability of Lipschitz mappings; Ramsey theory; embedding of metric spaces; Lipschitz mapping; differentiability of Lipschitz mappings; distortion; finite metric space
UR - http://eudml.org/doc/247391
ER -

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