16328721
Isoperimetric and Concentration Inequalities-Part I : Equivalence under Curvature Lower Bound
It is well known that isoperimetric inequalities imply in a very general measuremetric-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. As a corollary, we can recover and extend all previously known (dimension dependent) results by generalizing an isoperimetric inequality of Bobkov, and provide a new proof that under natural convexity assumptions, arbitrarily weak concentration implies a dimension independent linear isoperimetric inequality. Further applications will be described in a subsequent work. 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.
117088211
The concentration of measure phenomenon
Concentration functions and inequalities Isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities Sharp bounds of Gaussian and empirical processes Selected applications References Index.
17294192
An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies
Abstract We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities proved by Bobkov, Ledoux [S.G. Bobkov, M. Ledoux, From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal. 10 (5) (2000) 1028–1052] and the isoperimetric inequalities due to Bakry, Ledoux [D. Bakry, M. Ledoux, Levy–Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (2) (1996) 259–281] and Bobkov, Zegarlinski [S.G. Bobkov, B. Zegarlinski, Entropy bounds and isoperimetry, Mem. Amer. Math. Soc. 176 (829) (2005), x+69]. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov, Milman [M. Gromov, V.D. Milman, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compos. Math. 62 (3) (1987) 263–282].
122460511
Concentration and isoperimetry are equivalent assuming curvature lower bound
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. We show that under a lower bound condition on the Bakry–Emery 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. To cite this article: E. Milman, C. R. Acad. Sci. Paris, Ser. I 347 (2009).
53955741
Mass Transport and Variants of the Logarithmic Sobolev Inequality
We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.
123049824
OPTIMAL INTEGRABILITY CONDITION FOR THE LOG-SOBOLEV INEQUALITY
AbstractLet M denote a connected complete Riemannian manifold (possibly with a convex boundary), ρ the Riemannian distance function from a fixed point and V ∈ C 2 (M) such that dμ V ≔ e V d x is a probability measure. For any K ≥ 0, we prove that K/2 is the infimum over all λ > 0 such that Ric M − Hess V ≥ −K and imply the log-Sobolev inequality for the Dirichlet form μ V (|∇ f | 2 ).
56405324
LOGARITHMIC SOBOLEV INEQUALITIES: CONDITIONS AND COUNTEREXAMPLES
Let M be any noncompact, connected, complete Riemannian manifold with Riemannian distance function (from a fixed point) . Consider L = + rV for some V 2 C 2 (M) with dµ := e V dx a probability measure. Define > 0 as the smallest possible constant such that for any K, " > 0, µ(exp((K + ") 2 )) K. It is shown in the paper that 2 ( 1 , 1 ).
120433074
Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator
Abstract. We establish, by simple semigroup arguments, a Lévy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature.
15844908
Spectral gap, logarithmic Sobolev constant, and geometric bounds
We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant. We examine similarly dimension free isoperimetric bounds using these parameters.
120121168
Logarithmic Sobolev inequalities on noncompact Riemannian manifolds
Summary. This paper presents a dimension-free Harnack type inequality for heat semigroups on manifolds, from which a dimension-free lower bound is obtained for the logarithmic Sobolev constant on compact manifolds and a new criterion is proved for the logarithmic Sobolev inequalities (abbrev. LSI) on noncompact manifolds. As a result, it is shown that LSI may hold even though the curvature of the operator is negative everywhere.
117013170
Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures
We discuss an approach, based on the Brunn–Minkowski inequality, to isoperimetric and analytic inequalities for probability measures on Euclidean space with logarithmically concave densities. In particular, we show that such measures have positive isoperimetric constants in the sense of Cheeger and thus always share Poincare-type inequalities. We then describe those log-concave measures which satisfy isoperimetric inequalities of Gaussian type. The results are precised in dimension 1.
119419670
Uniform tail-decay of Lipschitz functions implies Cheeger's isoperimetric inequality under convexity assumptions
We show that for convex domains in Euclidean space, Cheeger's isoperimetric inequality, spectral-gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a priori weakest uniform tail-decay of these functions, are all equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz'ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude the stability of the spectral-gap for convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization of the Cheeger constant, as one over the expectation of the distance from the “worst” Borel set having half the measure of the convex domain. In addition, we easily recover (and extend) many previously known lower bounds, due to Payne–Weinberger, Li–Yau and Kannan–Lovasz–Simonovits, on the Cheeger constant of convex domains. Essential to our proof is a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0,∞) curvature-dimension condition of Bakry–Emery. To cite this article: E. Milman, C. R. Acad. Sci. Paris, Ser. I 346 (2008).
15512270
On the role of convexity in isoperimetry, spectral gap and concentration
We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan–Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0,∞) curvature-dimension condition of Bakry-Émery.
15807464
Manifolds with Density
Introduction We consider a Riemannian manifold M n with a positive density function Ψ (x) used to weight volume and hypersurface area. In terms of the underlying Riemannian volume dV 0 and area dA 0 , the new, weighted volume and area are given by
117765973
Metric Structures for Riemannian and Non-Riemannian Spaces
Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.- Manifolds with Bounded Ricci Curvature.- Isoperimetric Inequalities and Amenability.- Morse Theory and Minimal Models.- Pinching and Collapse.
54809186
A note on the isoperimetric constant
Reference EPFL-ARTICLE-161409 URL: http://archive.numdam.org/article/ASENS_1982_4_15_2_213_0.pdf Record created on 2010-12-03, modified on 2017-05-12
119968388
On the cases of equality in Bobkov's inequality and Gaussian rearrangement
Abstract. We determine all of the cases of equality in a recent inequality of Bobkov that implies the isoperimetric inequality on Gauss space. As an application we determine all of the cases of equality in the Gauss space analog of the Faber-Krahn inequality.
118512030
Minimal surfaces and functions of bounded variation
I: Parametric Minimal Surfaces.- 1. Functions of Bounded Variation and Caccioppoli Sets.- 2. Traces of BV Functions.- 3. The Reduced Boundary.- 4. Regularity of the Reduced Boundary.- 5. Some Inequalities.- 6. Approximation of Minimal Sets (I).- 7. Approximation of Minimal Sets (II).- 8. Regularity of Minimal Surfaces.- 9. Minimal Cones.- 10. The First and Second Variation of the Area.- 11. The Dimension of the Singular Set.- II: Non-Parametric Minimal Surfaces.- 12. Classical Solutions of the Minimal Surface Equation.- 13. The a priori Estimate of the Gradient.- 14. Direct Methods.- 15. Boundary Regularity.- 16. A Further Extension of the Notion of Non-Parametric Minimal Surface.- 17. The Bernstein Problem.- Appendix A.- Appendix B.- Appendix C.
53678404
A general comparison theorem with applications to volume estimates for submanifolds
© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1978, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
54062817
Regularity of isoperimetric hypersurfaces in Riemannian manifolds
We add to the literature the well-known fact that an isoperimetric hypersurface S of dimension at most six in a smooth Riemannian manifold M is a smooth submanifold. If the metric is merely Lipschitz, then S is still Holder differentiable.
59816096
Minimal cones and the Bernstein problem
DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung dieses Dokuments. Dieses Dokument ist ausschließlich für den persönlichen, nicht kommerziellen Gebrauch bestimmt. Das Copyright bleibt bei den Herausgebern oder sonstigen Rechteinhabern. Als Nutzer sind Sie sind nicht dazu berechtigt, eine Lizenz zu übertragen, zu transferieren oder an Dritte weiter zu geben. Die Nutzung stellt keine Übertragung des Eigentumsrechts an diesem Dokument dar und gilt vorbehaltlich der folgenden Einschränkungen: Sie müssen auf sämtlichen Kopien dieses Dokuments alle Urheberrechtshinweise und sonstigen Hinweise auf gesetzlichen Schutz beibehalten; und Sie dürfen dieses Dokument nicht in irgend einer Weise abändern, noch dürfen Sie dieses Dokument für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, aufführen, vertreiben oder anderweitig nutzen; es sei denn, es liegt Ihnen eine schriftliche Genehmigung von DigiZeitschriften e.V. und vom Herausgeber oder sonstigen Rechteinhaber vor. Mit dem Gebrauch von DigiZeitschriften e.V. und der Verwendung dieses Dokuments erkennen Sie die Nutzungsbedingungen an.
119823658
Some Sharp Isoperimetric Theorems for Riemannian Manifolds
We prove that a region of small prescribed volume in a smooth, compact Riemannian manifold has at least as much perimeter as a round ball in the model space form, using dif- ferential inequalities and the Gauss-Bonnet-Chern theorem with boundary term. First we show that a minimizer is a nearly round sphere. We also provide some new isoperimetric inequalities in surfaces.
14253599
Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds
We prove that the isoperimetric profile of a convex domain Ω with compact closure in a Riemannian manifold (M n+1 , g) satisfies a second-order differential inequality that only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of Q. Regularity properties of the profile and topological consequences on isoperimetric regions arise naturally from this differential point of view. Moreover, by integrating the differential inequality, we obtain sharp comparison theorems: not only can we derive an inequality that should be compared with Levy-Gromov Inequality but we also show that if Ric > n6 on Q, then the profile of Q is bounded from above by the profile of the half-space H n+1 δ in the simply connected space form with constant sectional curvature δ. As a consequence of isoperimetric comparisons we obtain geometric estimations for the volume and the diameter of Ω, and for the first non-zero Neumann eigenvalue for the Laplace operator on Ω.
51821681
Extremal properties of half-spaces for log-concave distributions
n w x n where m s m = ??? = m is the product measure in R , D s y1, 1 is the n n-dimensional cube in R n and the infimum is then over all Borel-measurable n n . sets A ; R of measure m A G p; 0 - p - 1, h ) 0. We are searching for necessary and sufficent conditions under which the . n 4 infimum in 1.1 is attained at the half-spaces A s x g R : x F c of 1 measure p, for all p and h the half-spaces of this form will be called . standard . In these notes, we give such conditions in the case where m is log-concave. There exists the following probabilistic consequence of the above property. If z , . . . , z are independent random variables on some probability space 1 n . V, I, P , with common distribution m, consider a family of random variables . S t , indexed by a set T and formed by linear combinations, n S t s a t z , . . i i is1 where the coefficients a are arbitrary functions on T such that i n < < s s sup a t F 1. . 1 i t is1 . Set M s sup S t . Then the extremal property of the standard half-spaces in t
54846973
On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint
among all competitors F C Q such that \F\ = \E\ and such that ||XF — X^IILUQ) < ^ f some 6 > 0. We prove that when Q is convex, the boundary dE Pi O is connected, or else dE Pi O consists of parallel planes meeting 9fi orthogonally. The result arises as an application of a property we derive for normal variations of constant mean curvature hypersurfaces bounding sets within a convex domain Q. The property states that for such variations, area is a concave function of the enclosed volume. Our results hold in all dimensions n, even in the presence of singularities.
118106464
Note on the Isoperimetric Profile of a Convex Body
A solution to the relative isoperimetric problem (or partitioning problem) is a subset E of a given set Ω ⊂ ℝ n (the container) with prescribed volume |E| =V and minimal area \( \int_\Omega {\left| {D_{XE} } \right|\, = \,A(V)} \) of the interface. Improving a result due to Sternberg & Zumbrun [8], we obtain that if Ω is convex then \( A{(V)^{{\frac{n}{{n - 1}}}}} \) is a concave function of V. As consequence we deduce that the isoperimetric ratio of E is no worse than that of the half-ball contained in ℍ = ℝ n-1 X (0, ∞), and that the mean curvature of the minimizer is bounded a priori.
24175620
A functional form of the isoperimetric inequality for the Gaussian measure
Abstract Letgbe a smooth function on R nwith values in [0, 1]. Using the isoperimetric property of the Gaussian measure, itis proved thatφ(Φ−1(Eg))−Eφ(Φ−1(g))⩽E|∇g|. Conversely, this inequality implies the isoperimetric property of the Gaussian measure.