Properties of isoperimetric, functional and Transport-Entropy inequalities via concentration

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 \emph{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.

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

We establish lower estimates for an integral functional$$\int\limits_\Omega f(u(x), \nabla u(x)) \, dx ,$$where \(\Omega\) -- a bounded domain in \(\mathbb{R}^n \; (n \geqslant 2)\), an integrand \(f(t,p) \, (t \in [0, \infty),\; p \in \mathbb{R}^n)\) -- a function that is \(B\)-measurable with respect to a variable \(t\) and is convex and even in the variable \(p\), \(\nabla u(x)\) -- a gradient (in the sense of Sobolev) of the function \(u \colon \Omega \rightarrow \mathbb{R}\). In the first and the second sections we utilize properties of permutations of differentiable functions and an isoperimetric inequality \(H^{n-1}( \partial A) \geqslant \lambda(m_n A)\), that connects \((n-1)\)-dimensional Hausdorff measure \(H^{n-1}(\partial A )\) of relative boundary \(\partial A\) of the set \(A \subset \Omega\) with its \(n\)-dimensional Lebesgue measure \(m_n A\). The integrand \(f\) is assumed to be isotropic, i.e. \(f(t,p) = f(t,q)\) if \(|p| = |q|\).Applications of the established results to multidimensional variational problems are outlined. For functions \( u \) that vanish on the boundary of the domain \(\Omega\), the assumption of the isotropy of the integrand \( f \) can be omitted. In this case, an important role is played by the Steiner and Schwartz symmetrization operations of the integrand \( f \) and of the function \( u \). The corresponding variants of the lower estimates are discussed in the third section. What is fundamentally new here is that the symmetrization operation is applied not only to the function \(u\), but also to the integrand \(f\). The geometric basis of the results of the third section is the Brunn-Minkowski inequality, as well as the symmetrization properties of the algebraic sum of sets.

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.

The Brunn-Minkowski and Prekopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions, so it seems that convexity has no special significance. On the other hand, it was recently shown that the Brunn-Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. In addition, a number of other geometric inequalities for convex sets arising from Brunn's concavity principle have recently been shown to yield local stochastic formulations, e.g., the Blaschke-Santalo inequality. In the first part of this dissertation, we study reverse inequalities for functionals of polar convex bodies invariant under the general linear group. We strengthen planar isoperimetric inequalities; we do that by attaching a stochastic model to some classical ones, such as Mahler's Theorem, and a reverse Lutwak-Zhang inequality, for the polar of L[subscript p] centroid bodies. In particular, we obtain the dual counterpart to a result of Bisztriczky-Boroczky. For the rest of the dissertation, we initiate a systematic study of stochastic isoperimetric inequalities for random functions. We show that for the subclass of log-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prekopa-Leindler inequality. Ultimately, we extend the latter result by developing stochastic geometry of s-concave functions. In this way, we establish local versions of dimensional forms of Brunn's principle a la Borell, Brascamp-Lieb, and Rinott. To do so, we define shadow systems of sconcave functions and revisit Rinott's approach in the context of multiple integral rearrangement inequalities.

The classical isoperimetric inequality in Euclidean space. Three different approaches.- The curve shortening flow and isoperimetric inequalities on surfaces.- $H^k$-flows and isoperimetric inequalities.- Estimates on the Willmore functional and isoperimetric inequalities.- Singularities in the volume-preserving mean curvature flow.- Bounds on the Heegaard genus of a hyperbolic manifold.- The isoperimetric profile for small volumes.- Local existence of flows driven by the second fundamental form and formation of singularities.- Invariance properties.- Singular behaviour of convex surfaces.- Convexity estimates.- Rescaling near a singularity.- Cylindrical and gradient estimates.- Mean curvature flow with surgeries.

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.

Affine vs. Euclidean isoperimetric inequalities

Continuing the development of a previous paper on generalized isoperimetric inequalities (i.e., rearrangement inequalities for Green's functions), we extend the theory to the case of Green's functions for a potential which approaches zero at infinity. Specialization to domain potentials and long times gives Pólya and Szegö's isoperimetric inequality for the electrostatic capacity. Long times and a more general potential give a new isoperimetric inequality (for the ``scattering length'' of a potential). We also obtain from another specialization a curious isoperimetric inequality for the trace of the phase shift operator of scattering theory (for a given energy).

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 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) .

Model spaces for sharp isoperimetric inequalities

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which are extremal for the isoperimetric problem. In particular, we recover the Gromov–Lévy and Bakry–Ledoux isoperimetric inequalities, which state that whenever the curvature is strictly positively bounded from below, these model spaces are the n -sphere and Gauss space, corresponding to generalized dimension being n and \infty , respectively. In all other cases, which seem new even for the classical Riemannian-volume measure, it turns out that there is no single model space to compare to, and that a simultaneous comparison to a natural one-parameter family of model spaces is required, nevertheless yielding a sharp result.

Concentration inequality around the thermal equilibrium measure of Coulomb gases