It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry–Émery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension independent bounds. Our method is entirely geometric, continuing the approach of Gromov, Buser and Morgan.
Il est connu que les inégalités isopérimétriques impliquent dans un cadre espace-mesure-métrique très général des inégalités de concentration correspondantes. Les premières bornent la mesure de bord d'un ensemble en fonction de sa mesure, tandis que les dernières bornent la mesure d'un ensemble qui est séparé d'un ensemble ayant la moitié de la mesure totale, en fonction de leur distance mutuelle. Nous montrons que sous une condition de borne inférieure sur le tenseur de courbure de Bakry–Émery d'une variété riemannienne équipée d'une densité, les inégalités de concentration complètement générales impliquent inversement leurs analogues isopérimétriques, à des constantes près qui sont indépendantes de la dimension. Notre méthode est entièrement géométrique, continuant l'approche de Gromov, Buser et Morgan.
Accepted:
Published online:
Emanuel Milman 1
@article{CRMATH_2009__347_1-2_73_0, author = {Emanuel Milman}, title = {Concentration and isoperimetry are equivalent assuming curvature lower bound}, journal = {Comptes Rendus. Math\'ematique}, pages = {73--76}, publisher = {Elsevier}, volume = {347}, number = {1-2}, year = {2009}, doi = {10.1016/j.crma.2008.11.014}, language = {en}, }
Emanuel Milman. Concentration and isoperimetry are equivalent assuming curvature lower bound. Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 73-76. doi : 10.1016/j.crma.2008.11.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.11.014/
[1] Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177-206
[2] Lévy–Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math., Volume 123 (1996) no. 2, pp. 259-281
[3] Levels of concentration between exponential and Gaussian, Ann. Fac. Sci. Toulouse Math. (6), Volume 10 (2001) no. 3, pp. 393-404
[4] Mass transport and variants of the logarithmic Sobolev inequality, J. Geom. Anal., Volume 18 (2008) no. 4, pp. 921-979 | arXiv
[5] V. Bayle, Propriétés de concavité du profil isopérimétrique et applications, PhD thesis, Institut Joseph Fourier, Grenoble, 2004
[6] Isoperimetric and analytic inequalities for log-concave probability measures, Ann. Probab., Volume 27 (1999) no. 4, pp. 1903-1921
[7] A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4), Volume 15 (1982) no. 2, pp. 213-230
[8] Optimal integrability condition for the log-Sobolev inequality, Quart. J. Math., Volume 58 (2007) no. 1, pp. 17-22
[9] M. Gromov, Paul Lévy isoperimetric inequality, preprint, I.H.E.S., 1980
[10] The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001
[11] Spectral gap, logarithmic Sobolev constant, and geometric bounds, Surveys in Differential Geometry, vol. IX, Int. Press, Somerville, MA, 2004, pp. 219-240
[12] E. Milman, A geometric approach to isoperimetric inequalities, manuscript, 2008
[13] On the role of convexity in isoperimetry, spectral-gap and concentration, 2008 (submitted for publication) | arXiv
[14] Uniform tail-decay of Lipschitz functions implies Cheeger's isoperimetric inequality under convexity assumptions, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 346 (2008), pp. 989-994
[15] An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies, J. Funct. Anal., Volume 254 (2008) no. 5, pp. 1235-1268 | arXiv
[16] Manifolds with density, Notices Amer. Math. Soc., Volume 52 (2005) no. 8, pp. 853-858
[17] Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields, Volume 109 (1997) no. 3, pp. 417-424
[18] Logarithmic Sobolev inequalities: conditions and counterexamples, J. Operator Theory, Volume 46 (2001) no. 1, pp. 183-197
Cited by Sources:
Comments - Policy