リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

大学・研究所にある論文を検索できる 「EXPONENTIAL CONCENTRATION IN TERMS OF GROMOV–LEDOUX’S EXPANSION COEFFICIENTS ON A METRIC MEASURE SPACE AND ITS UPPER DIAMETER BOUND SATISFYING VOLUME DOUBLING」の論文概要。リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

コピーが完了しました

URLをコピーしました

論文の公開元へ論文の公開元へ
書き出し

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 大阪大学 DOI:10.18910/89327

2022.10

概要

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.

参考文献

[1] L. Ambrosio and P. Tilli: Topics on Analysis in Metric Spaces, Oxford Lecture Series in Mathematics and its Applications 25, Oxford University Press, Oxford, 2004.

[2] D. Amir and V.D. Milman: Unconditional and symmetric sets in n-dimensional normed spaces, Israel J. Math. 37 (1980), 3–20.

[3] D. Bakry and M. Ledoux: Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator, Duke Math. J. 85 (1996), 253–270.

[4] F. Barthe: Levels of concentration between exponential and Gaussian, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), 393–404.

[5] P. Be´rard and D. Meyer: Ine´galite´s isope´rime´triques et applications, Ann. Sci. E´ cole Norm. Supe´r. (4) 15(1982), 513–541.

[6] M. Berger: Encounter with a Geometer, Part II, Notices Amer. Math. Soc. 47 (2000), 326–340.

[7] J. Cheeger: A lower bound for the smallest eigenvalue of the Laplacian; in Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press, Princeton, New Jersey, 1970, 195–199.

[8] J. Cheeger: Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428–517.

[9] S.Y. Cheng: Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), 289– 297.

[10] M. Gromov: Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics 152, Birkha¨user Boston, Inc., Boston, MA, 1999. Based on the 1981 French original: Structures Me´triques des Varie´te´s Riemanniennes. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates.

[11] M. Gromov and V.D. Milman: A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), 843–854.

[12] S. Keith: A differentiable structure for metric measure spaces, Adv. Math. 183 (2004), 271–315.

[13] M. Ledoux: A remark on hypercontractivity and the concentration of measure phenomenon in a compact Riemannian manifold, Israel J. Math. 69 (1990), 361–370.

[14] M. Ledoux: Isoperimetry and Gaussian analysis; in Lectures on Probability Theory and Statistics, Ecole d’Ete´ de Probabilite´s de Saint-Flour XXIV–1994, Lecture Notes in Math. 1648, 165–294, Springer-Verlag, Berlin-Heidelberg, 1996.

[15] M. Ledoux: Concentration of measure and logarithmic Sobolev inequalities; in Seminaire de Probabilite´s XXXIII, Lecture Notes in Math. 1709, 120–216, Springer-Verlag, Berlin-Heidelberg, 1999.

[16] M. Ledoux: The Concentration of Measure Phenomenon, Math. Surveys Monogr. 89, Amer. Math. Soc., Providence, Rhode Island, 2001.

[17] M. Ledoux: Spectral gap, logarithmic Sobolev constant, and geometric bounds; in Eigenvalues of Lapla- cians and other geometric operators, Surv. Differ. Geom. 9 (2004), 219–240.

[18] M. Ledoux and M. Talagrand: Probability in Banach Spaces: Isoperimetry and Processes, Springer-Verlag, Berlin-Heidelberg, 1991.

[19] P. Li and S.-T. Yau: Estimates of eigenvalues of a compact Riemannian manifold; in Geometry of the Laplace operator, Proc. Sympos. Pure Math. 36, Amer. Math. Soc., Providence, Rhode Island, 1980, 205– 239.

[20] A. Lichnerowicz: Ge´ome´trie des groupes de transformations, Travaux et Recherches Mathe´matiques III, Dunod, Paris, 1958.

[21] E. Milman: On the role of convexity in isoperimetry, spectral gap and concentration, Invent. Math. 177 (2009), 1–43.

[22] S.B. Myers: Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401–404.

[23] A. Naor, Y. Rabani and A. Sinclair: Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs, J. Funct. Anal. 227 (2005), 273–303.

[24] T. Shioya: Metric measure geometry, Gromov’s theory of convergence and concentration of metrics and measures, IRMA Lectures in Mathematics and Theoretical Physics 25, EMS Publishing House, Zu¨rich, 2016.

[25] C. Villani: Optimal transport, Old and new, Grundlehren der mathematischen Wissenschaften 338, Springer-Verlag, Berlin Heidelberg, 2009.

[26] D. Yang: Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature, Pacific J. Math. 190 (1999), 383–398.

参考文献をもっと見る

全国の大学の
卒論・修論・学位論文

一発検索!

この論文の関連論文を見る