Isoperimetric and concentration inequalities: Equivalence under curvature lower bound

Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided L ∞ bound on the ratio between their densities, Wasserstein distances, and Kullback–Leibler divergence. In particular, an extension of the Holley–Stroock perturbation lemma for the log-Sobolev inequality is obtained, and the dependence on the perturbation parameter is improved from linear to logarithmic. Second, the equivalence of Transport-Entropy inequalities with different cost-functions is verified, by obtaining a reverse Jensen type inequality. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting. Of independent interest is a new dimension independent characterization of Transport-Entropy inequalities with respect to the 1-Wasserstein distance, which does not assume any curvature lower bound.

We introduce some methods to study the isoperimetric problem in 3-dimensional Riemannian manifolds and we show that in the positive curva- ture case we can control the topology of the isoperimetric regions. We consider the case of the projective space, which was first solved by Ritorea nd Ros, and we apply it to the Willmore problem. The isoperimetric problem is a classical topic in geometry but at the same time many basic questions about it remain unsolved. In this paper we first introduce some of the methods used in the study of that problem, although we will not try to be exhaustive at all. We will explain some relatively flexible ideas, like symmetrization or stability, which can be adapted to a certain number of situations. As an example we will study the problem for radial metrics on the 3-sphere. Then we will show that in 3-manifolds with positive Ricci curvature the topol- ogy of the isoperimetric regions can be controled. In particular we will prove that, when the volume of the ambient space is large, any isoperimetric surface must be either an sphere or a torus. As consequence, we will solve the isoperimetric problem in the real projective space or, equivalently, the isoperimetric problem for antipodal invariant regions in the 3-sphere. That result was first obtained by Ritorea nd Ros (35), but here we will give a somewhat different proof. Finally, as application of the above results, we will solve the Willmore conjec- ture for tori in euclidean space which are symmetric with respect to a point. 2. The isoperimetric problem In this paper we will only consider the three dimensional case. Let M be a Riemaniann 3-dimensional manifold with or without boundary and volume V(M ) ∈ )0, ∞). Given a positive number v< V(M ), we want to study the compact surfaces Σ ⊂ M such that

By using optimal mass transport theory we prove a sharp isoperimetric inequality in $${\textsf {CD}} (0,N)$$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of n-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of Brendle (Comm Pure Appl Math 2021:13717, 2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

This is a continuation of our previous work [Preprint, 2008, http://arxiv.org/abs/0712.4092]. It is well known that various isoperimetric inequalities imply their functional ‘counterparts’, but in general this is not an equivalence. We show that under certain convexity assumptions (for example, for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz–Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz'ya and extended by Barthe–Cattiaux–Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, q-log-Sobolev inequalities (q ∈ [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarliński. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvature–dimension condition of Bakry–Émery.

In this first chapter, we present the isoperimetric inequalities which now appear as the crucial concept in the understanding of various concentration inequalities, tail behaviors and integrability theorems in Probability in Banach spaces. These inequalities often arise as the final and most elaborate forms of previous, weaker (but already efficient) inequalities which will be mentioned in their framework throughout the book. In these final forms however, the isoperimetric inequalities and associated concentration of measure phenomena provide the appropriate ideas for an in depth comprehension of some of the most important theorems of the theory.

Isoperimetric inequalities in graphs and surfaces

Infimum-convolution description of concentration properties of product probability measures, with applications

This paper studies the stability of isoperimetric inequalities under quasi-isometries between non-exceptional Riemann surfaces endowed with their Poincare metrics. This stability was proved by Kanai in the more general setting of Riemannian manifolds under the condition of positive injectivity radius. The present work proves the stability of the linear isoperimetric inequality for planar surfaces (genus zero surfaces) without any condition on their injectivity radii. It is also shown the stability of any non-linear isoperimetric inequality.

Let Σ be a k-dimensional complete proper minimal submanifold in the Poincaré ball model Bn of hyperbolic geometry. If we consider Σ as a subset of the unit ball Bn in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold Σ and the ideal boundary ∂ ∞ Σ $\partial _\infty \Sigma $ , say Vol ℝ ( Σ ) $\operatorname{Vol}_{\mathbb {R}}(\Sigma )$ and Vol ℝ ( ∂ ∞ Σ ) $\operatorname{Vol}_{\mathbb {R}}(\partial _\infty \Sigma )$ , respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if Vol ℝ ( ∂ ∞ Σ ) ≥ Vol ℝ ( 𝕊 k - 1 ) $\operatorname{Vol}_{\mathbb {R}}(\partial _\infty \Sigma ) \ge \operatorname{Vol}_{\mathbb {R}}(\mathbb {S}^{k-1})$ , then Σ satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such Σ, we further obtain a sharp lower bound for the Euclidean volume Vol ℝ ( Σ ) $\operatorname{Vol}_{\mathbb {R}}(\Sigma )$ , which is an extension of Fraser–Schoen and Brendle's recent results to hyperbolic space. Moreover we introduce the Möbius volume of Σ in Bn to prove an isoperimetric inequality via the Möbius volume for Σ.

Concentration and isoperimetry are equivalent assuming curvature lower bound

We study the isoperimetric, functional and concentration properties of n n -dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension N N is negative and, more generally, is in the range N ∈ ( − ∞ , 1 ) N \in (-\infty ,1) , extending the scope from the traditional range N ∈ [ n , ∞ ] N \in [n,\infty ] . In particular, we identify the correct one-dimensional model-spaces under an additional diameter upper bound and discover a new case yielding a single model space (besides the previously known N N -sphere and Gaussian measure when N ∈ [ n , ∞ ] N \in [n,\infty ] ): a (positively curved) sphere of (possibly negative) dimension N ∈ ( − ∞ , 1 ) N \in (-\infty ,1) . When curvature is non-negative, we show that arbitrarily weak concentration implies an N N -dimensional Cheeger isoperimetric inequality and derive various weak Sobolev and Nash-type inequalities on such spaces. When curvature is strictly positive, we observe that such spaces satisfy a Poincaré inequality uniformly for all N ∈ ( − ∞ , 1 − ε ] N \in (-\infty ,1-\varepsilon ] and enjoy a two-level concentration of the type exp ⁡ ( − min ( t , t 2 ) ) \exp (-\min (t,t^2)) . Our main technical tool is a generalized version of the Heintze–Karcher theorem, which we extend to the range N ∈ ( − ∞ , 1 ) N \in (-\infty ,1) .

The generalized orthogonal ensemble satisfies isoperimetric inequalities analogous to the Gaussian isoperimetric inequality, and an analogue of Wigner's law. Let v be a continuous and even real function such that V(X)=tracev(X)/n defines a uniformly p-convex function on the real symmetric n×n matrices X for some p≥2. Then ν(dX)=e−V(X)dX/Z satisfies deviation and transportation inequalities analogous to those satisfied by Gaussian measure(6, 27), but for the Schatten cp norm. The map, that associates to each X∈Msn(ℝ) its ordered eigenvalue sequence, induces from ν a measure which satisfies similar inequalities. It follows from such concentration inequalities that the empirical distribution of eigenvalues converges weakly almost surely to some non-random compactly supported probability distribution as n→∞.

We develop several applications to almost sure limit theorems for sums of independent vector valued random variables of an isoperimetric inequality due to Talagrand. A general treatment of the classical laws of large numbers of Kolmogorov and Prokorov and laws of the iterated logarithm of Kolmogorov and Hartman and Wintner is described. New results as well as simpler new proofs of known ones illustrate the usefulness of isoperimetric methods in this context. We show further how this approach can be used in the study of limit theorems for trimmed sums of independent and identically distributed random variables.

We study in this paper the relationship of isoperimetric inequality and hyperbolicity for graphs and Riemannian manifolds. We obtain a characterization of graphs and Riemannian manifolds (with bounded local geometry) satisfying the (Cheeger) isoperimetric inequality, in terms of their Gromov boundary, improving similar results from a previous work. In particular, we prove that having a pole is a necessary condition to have isoperimetric inequality and, therefore, it can be removed as hypothesis.