EXPONENTIAL CONCENTRATION IN TERMS OF GROMOV–LEDOUX’S EXPANSION COEFFICIENTS ON A METRIC MEASURE SPACE AND ITS UPPER DIAMETER BOUND SATISFYING VOLUME DOUBLING

Tanaka, Ushio

doi:https://doi.org/10.18910/89327

EXPONENTIAL CONCENTRATION IN TERMS OF GROMOV–LEDOUX’S EXPANSION COEFFICIENTS ON A METRIC MEASURE SPACE AND ITS UPPER DIAMETER BOUND SATISFYING VOLUME DOUBLING

Tanaka, Ushio

Osaka Journal of Mathematics, 2022, 59(4), 727-753

アクセス数:251件（2026-02-07 18:19 集計）

固定URL: https://doi.org/10.18910/89327

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タイトル	EXPONENTIAL CONCENTRATION IN TERMS OF GROMOV–LEDOUX’S EXPANSION COEFFICIENTS ON A METRIC MEASURE SPACE AND ITS UPPER DIAMETER BOUND SATISFYING VOLUME DOUBLING
著者	NRID 1000060516897 1000060516897 著者名 Tanaka, Ushio
抄録	To investigate a concentration of measure phenomena on metric measure spaces in terms of Gromov–Ledoux’s expansion coefficients on this space as well as Ledoux’s per se, we studied a concentration function in concert with their expansion coefficients. Further investigation into an exponential concentration in terms of Ledoux’s expansion coefficient on a bounded and volume doubling metric measure space enables us to derive an upper bound for its diameter in terms of both the Ledoux’s expansion coefficient and doubling constant, provided that Ledoux’s expansion coefficient > 1. In this study, we let Ledoux’s expansion coefficient > 1 on a metric measure space, which is ensured by adopting Poincare inequality. We demonstrated that on a metric measure space, Gromov–Ledoux’s expansion coefficients with Ledoux’s expansion coefficient > 1 give rise to an exponential concentration in terms of themselves. We further showed that on a bounded and volume doubling metric measure space, a Ledoux’s expansion coefficient of order bounded from above in terms of both the doubling constant > 1 and its diameter is bounded from above in terms of the doubling constant per se. We applied this upper diameter bound to a closed smooth Riemannian manifold with non-negative Ricci curvature. This upper bound is described in terms of both the spectral gap and dimension.
出版者	Osaka University and Osaka Metropolitan University, Departments of Mathematics
収録物名	Osaka Journal of Mathematics
巻(号)	59(4)
ページ	727-753
刊行年月	2022-10
言語	英語
ハンドルURL	https://hdl.handle.net/11094/89327
DOI	https://doi.org/10.18910/89327
PISSN	00306126
NCID	AA00765910
アクセス権	open access
出版タイプ	Version of Record
カテゴリ	紀要論文 Departmental Bulletin Paper
カテゴリ	Osaka Journal of Mathematics / Volume 59, Number 4 （2022-10）

資源タイプ	紀要論文
ローカル資源タイプ	紀要論文
DCMI資源タイプ	text
DCTERMS.bibliographicCitation	Osaka Journal of Mathematics. 2022, 59(4), p. 727-753
DC.title	EXPONENTIAL CONCENTRATION IN TERMS OF GROMOV–LEDOUX’S EXPANSION COEFFICIENTS ON A METRIC MEASURE SPACE AND ITS UPPER DIAMETER BOUND SATISFYING VOLUME DOUBLING
DC.creator	Tanaka, Ushio
DC.publisher	Osaka University and Osaka Metropolitan University, Departments of Mathematics
DC.language' scheme='DCTERMS.RFC1766	英語
DC.type' scheme='DCTERMS.DCMIType	text
DCTERMS.issued' scheme='DCTERMS.W3CDTF	2022-10
DCTERMS.abstract	To investigate a concentration of measure phenomena on metric measure spaces in terms of Gromov–Ledoux’s expansion coefficients on this space as well as Ledoux’s per se, we studied a concentration function in concert with their expansion coefficients. Further investigation into an exponential concentration in terms of Ledoux’s expansion coefficient on a bounded and volume doubling metric measure space enables us to derive an upper bound for its diameter in terms of both the Ledoux’s expansion coefficient and doubling constant, provided that Ledoux’s expansion coefficient > 1. In this study, we let Ledoux’s expansion coefficient > 1 on a metric measure space, which is ensured by adopting Poincare inequality. We demonstrated that on a metric measure space, Gromov–Ledoux’s expansion coefficients with Ledoux’s expansion coefficient > 1 give rise to an exponential concentration in terms of themselves. We further showed that on a bounded and volume doubling metric measure space, a Ledoux’s expansion coefficient of order bounded from above in terms of both the doubling constant > 1 and its diameter is bounded from above in terms of the doubling constant per se. We applied this upper diameter bound to a closed smooth Riemannian manifold with non-negative Ricci curvature. This upper bound is described in terms of both the spectral gap and dimension.
DC.identifier	10.18910/89327
DC.format	89327
citation_title	EXPONENTIAL CONCENTRATION IN TERMS OF GROMOV–LEDOUX’S EXPANSION COEFFICIENTS ON A METRIC MEASURE SPACE AND ITS UPPER DIAMETER BOUND SATISFYING VOLUME DOUBLING
citation_author	Tanaka, Ushio
citation_publisher	Osaka University and Osaka Metropolitan University, Departments of Mathematics
citation_language	英語
citation_date	2022-10
citation_journal_title	Osaka Journal of Mathematics
citation_volume	59
citation_issue	4
citation_firstpage	727
citation_lastpage	753
citation_public_url	https://hdl.handle.net/11094/89327
citation_doi_detail	10.18910/89327
citation_doi	https://doi.org/10.18910/89327

EXPONENTIAL CONCENTRATION IN TERMS OF GROMOV–LEDOUX’S EXPANSION COEFFICIENTS ON A METRIC MEASURE SPACE AND ITS UPPER DIAMETER BOUND SATISFYING VOLUME DOUBLING

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