Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1

Abstract We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map %${u: X \rightarrow \mathbb R}%$ such that for every interval %${I \subset \mathbb R}%$, the connected components of u−1(I) have diameter %${\leq {...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Cheeger, Jeff [verfasserIn] Kleiner, Bruce

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2013

Schlagwörter:	Inverse Limit Isometric Embedding Inverse System Path Metrics Direction Preserve

Anmerkung:	© Springer Basel 2013

Übergeordnetes Werk:	Enthalten in: Geometric and functional analysis - Cham (ZG) : Springer International Publishing AG, 1991, 23(2013), 1 vom: 22. Jan., Seite 96-133
Übergeordnetes Werk:	volume:23 ; year:2013 ; number:1 ; day:22 ; month:01 ; pages:96-133

Links:	Volltext

DOI / URN:	10.1007/s00039-012-0201-8

Katalog-ID:	SPR00035497X

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520			\|a Abstract We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map %${u: X \rightarrow \mathbb R}%$ such that for every interval %${I \subset \mathbb R}%$, the connected components of u−1(I) have diameter %${\leq {\rm const} \cdot {\rm diam}(I)}%$, then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1.
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allfields	10.1007/s00039-012-0201-8 doi (DE-627)SPR00035497X (SPR)s00039-012-0201-8-e DE-627 ger DE-627 rakwb eng Cheeger, Jeff verfasserin aut Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1 2013 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer Basel 2013 Abstract We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map %${u: X \rightarrow \mathbb R}%$ such that for every interval %${I \subset \mathbb R}%$, the connected components of u−1(I) have diameter %${\leq {\rm const} \cdot {\rm diam}(I)}%$, then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1. Inverse Limit (dpeaa)DE-He213 Isometric Embedding (dpeaa)DE-He213 Inverse System (dpeaa)DE-He213 Path Metrics (dpeaa)DE-He213 Direction Preserve (dpeaa)DE-He213 Kleiner, Bruce aut Enthalten in Geometric and functional analysis Cham (ZG) : Springer International Publishing AG, 1991 23(2013), 1 vom: 22. Jan., Seite 96-133 (DE-627)25372368X (DE-600)1459155-8 1420-8970 nnns volume:23 year:2013 number:1 day:22 month:01 pages:96-133 https://dx.doi.org/10.1007/s00039-012-0201-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 23 2013 1 22 01 96-133
spelling	10.1007/s00039-012-0201-8 doi (DE-627)SPR00035497X (SPR)s00039-012-0201-8-e DE-627 ger DE-627 rakwb eng Cheeger, Jeff verfasserin aut Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1 2013 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer Basel 2013 Abstract We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map %${u: X \rightarrow \mathbb R}%$ such that for every interval %${I \subset \mathbb R}%$, the connected components of u−1(I) have diameter %${\leq {\rm const} \cdot {\rm diam}(I)}%$, then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1. Inverse Limit (dpeaa)DE-He213 Isometric Embedding (dpeaa)DE-He213 Inverse System (dpeaa)DE-He213 Path Metrics (dpeaa)DE-He213 Direction Preserve (dpeaa)DE-He213 Kleiner, Bruce aut Enthalten in Geometric and functional analysis Cham (ZG) : Springer International Publishing AG, 1991 23(2013), 1 vom: 22. Jan., Seite 96-133 (DE-627)25372368X (DE-600)1459155-8 1420-8970 nnns volume:23 year:2013 number:1 day:22 month:01 pages:96-133 https://dx.doi.org/10.1007/s00039-012-0201-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 23 2013 1 22 01 96-133
allfields_unstemmed	10.1007/s00039-012-0201-8 doi (DE-627)SPR00035497X (SPR)s00039-012-0201-8-e DE-627 ger DE-627 rakwb eng Cheeger, Jeff verfasserin aut Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1 2013 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer Basel 2013 Abstract We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map %${u: X \rightarrow \mathbb R}%$ such that for every interval %${I \subset \mathbb R}%$, the connected components of u−1(I) have diameter %${\leq {\rm const} \cdot {\rm diam}(I)}%$, then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1. Inverse Limit (dpeaa)DE-He213 Isometric Embedding (dpeaa)DE-He213 Inverse System (dpeaa)DE-He213 Path Metrics (dpeaa)DE-He213 Direction Preserve (dpeaa)DE-He213 Kleiner, Bruce aut Enthalten in Geometric and functional analysis Cham (ZG) : Springer International Publishing AG, 1991 23(2013), 1 vom: 22. Jan., Seite 96-133 (DE-627)25372368X (DE-600)1459155-8 1420-8970 nnns volume:23 year:2013 number:1 day:22 month:01 pages:96-133 https://dx.doi.org/10.1007/s00039-012-0201-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 23 2013 1 22 01 96-133
allfieldsGer	10.1007/s00039-012-0201-8 doi (DE-627)SPR00035497X (SPR)s00039-012-0201-8-e DE-627 ger DE-627 rakwb eng Cheeger, Jeff verfasserin aut Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1 2013 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer Basel 2013 Abstract We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map %${u: X \rightarrow \mathbb R}%$ such that for every interval %${I \subset \mathbb R}%$, the connected components of u−1(I) have diameter %${\leq {\rm const} \cdot {\rm diam}(I)}%$, then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1. Inverse Limit (dpeaa)DE-He213 Isometric Embedding (dpeaa)DE-He213 Inverse System (dpeaa)DE-He213 Path Metrics (dpeaa)DE-He213 Direction Preserve (dpeaa)DE-He213 Kleiner, Bruce aut Enthalten in Geometric and functional analysis Cham (ZG) : Springer International Publishing AG, 1991 23(2013), 1 vom: 22. Jan., Seite 96-133 (DE-627)25372368X (DE-600)1459155-8 1420-8970 nnns volume:23 year:2013 number:1 day:22 month:01 pages:96-133 https://dx.doi.org/10.1007/s00039-012-0201-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 23 2013 1 22 01 96-133
allfieldsSound	10.1007/s00039-012-0201-8 doi (DE-627)SPR00035497X (SPR)s00039-012-0201-8-e DE-627 ger DE-627 rakwb eng Cheeger, Jeff verfasserin aut Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1 2013 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer Basel 2013 Abstract We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map %${u: X \rightarrow \mathbb R}%$ such that for every interval %${I \subset \mathbb R}%$, the connected components of u−1(I) have diameter %${\leq {\rm const} \cdot {\rm diam}(I)}%$, then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1. Inverse Limit (dpeaa)DE-He213 Isometric Embedding (dpeaa)DE-He213 Inverse System (dpeaa)DE-He213 Path Metrics (dpeaa)DE-He213 Direction Preserve (dpeaa)DE-He213 Kleiner, Bruce aut Enthalten in Geometric and functional analysis Cham (ZG) : Springer International Publishing AG, 1991 23(2013), 1 vom: 22. Jan., Seite 96-133 (DE-627)25372368X (DE-600)1459155-8 1420-8970 nnns volume:23 year:2013 number:1 day:22 month:01 pages:96-133 https://dx.doi.org/10.1007/s00039-012-0201-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 23 2013 1 22 01 96-133
language	English
source	Enthalten in Geometric and functional analysis 23(2013), 1 vom: 22. Jan., Seite 96-133 volume:23 year:2013 number:1 day:22 month:01 pages:96-133
sourceStr	Enthalten in Geometric and functional analysis 23(2013), 1 vom: 22. Jan., Seite 96-133 volume:23 year:2013 number:1 day:22 month:01 pages:96-133
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Inverse Limit Isometric Embedding Inverse System Path Metrics Direction Preserve
isfreeaccess_bool	false
container_title	Geometric and functional analysis
authorswithroles_txt_mv	Cheeger, Jeff @@aut@@ Kleiner, Bruce @@aut@@
publishDateDaySort_date	2013-01-22T00:00:00Z
hierarchy_top_id	25372368X
id	SPR00035497X
language_de	englisch
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author	Cheeger, Jeff
spellingShingle	Cheeger, Jeff misc Inverse Limit misc Isometric Embedding misc Inverse System misc Path Metrics misc Direction Preserve Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1
authorStr	Cheeger, Jeff
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topic_title	Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1 Inverse Limit (dpeaa)DE-He213 Isometric Embedding (dpeaa)DE-He213 Inverse System (dpeaa)DE-He213 Path Metrics (dpeaa)DE-He213 Direction Preserve (dpeaa)DE-He213
topic	misc Inverse Limit misc Isometric Embedding misc Inverse System misc Path Metrics misc Direction Preserve
topic_unstemmed	misc Inverse Limit misc Isometric Embedding misc Inverse System misc Path Metrics misc Direction Preserve
topic_browse	misc Inverse Limit misc Isometric Embedding misc Inverse System misc Path Metrics misc Direction Preserve
format_facet	Elektronische Aufsätze Aufsätze Elektronische Ressource
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title	Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1
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title_full	Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1
author_sort	Cheeger, Jeff
journal	Geometric and functional analysis
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author_browse	Cheeger, Jeff Kleiner, Bruce
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author-letter	Cheeger, Jeff
doi_str_mv	10.1007/s00039-012-0201-8
title_sort	realization of metric spaces as inverse limits, and bilipschitz embedding in l1
title_auth	Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1
abstract	Abstract We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map %${u: X \rightarrow \mathbb R}%$ such that for every interval %${I \subset \mathbb R}%$, the connected components of u−1(I) have diameter %${\leq {\rm const} \cdot {\rm diam}(I)}%$, then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1. © Springer Basel 2013
abstractGer	Abstract We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map %${u: X \rightarrow \mathbb R}%$ such that for every interval %${I \subset \mathbb R}%$, the connected components of u−1(I) have diameter %${\leq {\rm const} \cdot {\rm diam}(I)}%$, then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1. © Springer Basel 2013
abstract_unstemmed	Abstract We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map %${u: X \rightarrow \mathbb R}%$ such that for every interval %${I \subset \mathbb R}%$, the connected components of u−1(I) have diameter %${\leq {\rm const} \cdot {\rm diam}(I)}%$, then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1. © Springer Basel 2013
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container_issue	1
title_short	Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1
url	https://dx.doi.org/10.1007/s00039-012-0201-8
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author2	Kleiner, Bruce
author2Str	Kleiner, Bruce
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doi_str	10.1007/s00039-012-0201-8
up_date	2024-07-03T15:31:14.551Z
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Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1

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