Duke Mathematical Journal

Isoperimetric and concentration inequalities: Equivalence under curvature lower bound

Emanuel Milman

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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. The reverse implication is in general false. It is shown that under a (possibly negative) lower bound condition on a natural notion of curvature associated to a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension independent bounds. The results are essentially the best possible (up to constants) and significantly extend all previously known results, which could deduce only dimension-dependent bounds or could not deduce anything stronger than a linear isoperimetric inequality in the restrictive nonnegative curvature setting. As a corollary, all of these previous results are recovered and extended by generalizing an isoperimetric inequality of Bobkov. Further applications will be described in subsequent works. Contrary to previous attempts in this direction, our method is entirely geometric, continuing the approach set forth by Gromov and adapted to the manifold-with-density setting by Morgan.

Article information

Source
Duke Math. J., Volume 154, Number 2 (2010), 207-239.

Dates
First available in Project Euclid: 16 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1281963649

Digital Object Identifier
doi:10.1215/00127094-2010-038

Mathematical Reviews number (MathSciNet)
MR2682183

Zentralblatt MATH identifier
1205.53038

Subjects
Primary: 32F32: Analytical consequences of geometric convexity (vanishing theorems, etc.)
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]

Citation

Milman, Emanuel. Isoperimetric and concentration inequalities: Equivalence under curvature lower bound. Duke Math. J. 154 (2010), no. 2, 207--239. doi:10.1215/00127094-2010-038. https://projecteuclid.org/euclid.dmj/1281963649


Export citation

References

  • D. Bakry and M. Émery, “Diffusions hypercontractives” in Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math. 1123, Springer, Berlin, 1985, 177–206.
  • D. Bakry and M. Ledoux, Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (1996), 259–281.
  • D. Bakry, M. Ledoux, and Z. Qian, Logarithmic Sobolev inequalities, Poincaré inequalities and heat kernel bounds, unpublished manuscript, 1997.
  • F. Barthe, Levels of concentration between exponential and Gaussian, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), 393–404.
  • F. Barthe and A. V. Kolesnikov, Mass transport and variants of the logarithmic Sobolev inequality, J. Geom. Anal. 18 (2008), 921–979.
  • C. Bavard and P. Pansu, Sur le volume minimal de $\bold R\sp 2$, Ann. Sci. École Norm. Sup. (4) 19 (1986), 479–490.
  • V. Bayle, Propriétés de concavité du profil isopérimétrique et applications, Ph.D. dissertation, Institut Joseph Fourier, Grenoble, 2004.
  • V. Bayle and C. Rosales, Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds, Indiana Univ. Math. J. 54 (2005), 1371–1394.
  • P. Bérard, G. Besson, and S. Gallot, Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov, Invent. Math. 80 (1985), 295–308.
  • S. G. Bobkov, A functional form of the isoperimetric inequality for the Gaussian measure, J. Funct. Anal. 135 (1996), 39–49.
  • —, Extremal properties of half-spaces for log-concave distributions, Ann. Probab. 24 (1996), 35–48.
  • —, Isoperimetric and analytic inequalities for log-concave probability measures, Ann. Probab. 27 (1999), 1903–1921.
  • —, A localized proof of the isoperimetric Bakry-Ledoux inequality and some applications, Teor. Veroyatnost. i Primenen. 47 (2002), 340–346.; English translation in Theory Probab. Appl. 47 (2003), 308–314.
  • E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268.
  • C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), 207–216.
  • Y. D. Burago and V. A. Zalgaller, Geometric Inequalities, Grundlehren Math. Wiss. 288, Springer, Berlin, 1988.
  • P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), 213–230.
  • E. A. Carlen and C. Kerce, On the cases of equality in Bobkov's inequality and Gaussian rearrangement, Calc. Var. Partial Differential Equations 13 (2001), 1–18.
  • X. Chen and F.-Y. Wang, Optimal integrability condition for the log-Sobolev inequality, Q. J. Math. 58 (2007), 17–22.
  • A. Ehrhard, Éléments extrémaux pour les inégalités de Brunn-Minkowski gaussiennes, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), 149–168.
  • H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York, 1969.
  • S. Gallot, “Inégalités isopérimétriques et analytiques sur les variétés riemanniennes” in On the Geometry of Differentiable Manifolds (Rome, 1986), Astérisque 163 –164., Soc. Math. France, Montrouge, 1988, 31–91.
  • S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, 3rd ed., Universitext, Springer, Berlin, 2004.
  • E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. 80, Birkhäuser, Basel, 1984.
  • M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, with appendices by M. Katz, P. Pansu, and S. Semmes, Progr. Math. 152, Birkhäuser, Boston, 1999.
  • —, Paul Lévy's isoperimetric inequality, preprint, 1980.
  • M. Grüter, Boundary regularity for solutions of a partitioning problem, Arch. Rational Mech. Anal. 97 (1987), 261–270.
  • E. Heintze and H. Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. École Norm. Sup. (4) 11 (1978), 451–470.
  • E. Kuwert, “Note on the isoperimetric profile of a convex body” in Geometric Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2003, 195–200.
  • M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monogr. 89, Amer. Math. Soc., Providence, 2001.
  • —, “Spectral gap, logarithmic Sobolev constant, and geometric bounds” in Surveys in Differential Geometry, Vol. IX, Surv. Differ. Geom. IX, Int. Press, Somerville, Mass., 2004, 219–240.
  • M. Lee, Isoperimetric regions in spaces, Bachelor's degree thesis, Williams College, Williamstown, Mass., 2006.
  • E. Milman, Uniform tail-decay of Lipschitz functions implies Cheeger's isoperimetric inequality under convexity assumptions, C. R. Math. Acad. Sci. Paris 346 (2008), 989–994.
  • —, Concentration and isoperimetry are equivalent assuming curvature lower bound, C. R. Math. Acad. Sci. Paris 347 (2009), 73–76.
  • —, On the role of convexity in functional and isoperimetric inequalities, Proc. Lond. Math. Soc. (3) 99 (2009), 32–66.
  • —, On the role of convexity in isoperimetry, spectral gap and concentration, Invent. Math. 177 (2009), 1–43.
  • —, Properties of isoperimetric, functional and transport-entropy inequalities via concentration, preprint.
  • —, Optimal isoperimetric and Sobolev inequalities on compact manifolds with density, in preparation.
  • E. Milman and S. Sodin, An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies, J. Funct. Anal. 254 (2008), 1235–1268.
  • V. D. Milman, “The heritage of P. Lévy in geometrical functional analysis” in Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987), Astérisque 157 –158., Soc. Math. France, Montrouge, 1988, 273–301.
  • F. Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc. 355 (2003), 5041–5052.
  • —, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853–858.
  • —, Geometric Measure Theory: A Beginner's Guide, 4th ed., Elsevier/Academic Press, Amsterdam, 2009.
  • F. Morgan and D. L. Johnson, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana Univ. Math. J. 49 (2000), 1017–1041.
  • P. Sternberg and K. Zumbrun, On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint, Comm. Anal. Geom. 7 (1999), 199–220.
  • V. N. Sudakov and B. S. Cirel'Son [Tsirelson], “Extremal properties of half-spaces for spherically invariant measures” in Problems in the Theory of Probability Distributions, II, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41, Leningrad, 1974, 14–24.
  • F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997), 417–424.
  • —, Logarithmic Sobolev inequalities: Conditions and counterexamples, J. Operator Theory 46 (2001), 183–197.