Isoperimetric and Functional Inequalities

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.

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.

For each natural number n n and any bounded, convex domain Ω ⊂ R n \Omega \subset \mathbb {R}^n we characterize the sharp constant C ( n , Ω ) C(n,\Omega ) in the Poincaré inequality ‖ f − f ¯ Ω ‖ L ∞ ( Ω ; R ) ≤ C ( n , Ω ) ‖ ∇ f ‖ L ∞ ( Ω ; R ) \| f - \bar {f}_{\Omega }\|_{L^{\infty }(\Omega ;\mathbb {R})} \leq C(n,\Omega ) \|\nabla f\|_{L^{\infty }(\Omega ;\mathbb {R})} . Here, f ¯ Ω \bar {f}_{\Omega } denotes the mean value of f f over Ω \Omega . In the case that Ω \Omega is a ball B r B_r of radius r r in R n \mathbb {R}^n , we calculate C ( n , B r ) = C ( n ) r C(n,B_r)=C(n)r explicitly in terms of n n and a ratio of the volumes of the unit balls in R 2 n − 1 \mathbb {R}^{2n-1} and R n \mathbb {R}^n . More generally, we prove that C ( n , B r ( Ω ) ) ≤ C ( n , Ω ) ≤ n n + 1 d i a m ( Ω ) C(n,B_{r(\Omega )}) \leq C(n,\Omega ) \leq \frac {n}{n+1}\mathrm {diam}(\Omega ) , where B r ( Ω ) B_{r(\Omega )} is a ball in R n \mathbb {R}^n with the same n − n- dimensional Lebesgue measure as Ω \Omega . Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant.

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 Poincaré-type inequalities. We then describe those log-concave measures which satisfy isoperimetric inequalities of Gaussian type. The results are precised in dimension 1.

The Brunn–Minkowski inequality for volume differences

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.

A method of symmetrizing functions and its application to certain problems in elasticity theory for non-uniform bodies

In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of Rn, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

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).

After Hormander's fundamental paper on hypoellipticity [54], the study of partial differential equations arising from families of noncommuting vector fields has developed significantly. In this paper we study some basic functional and geometric properties of general families of vector fields that include the Hormander type as a special case. Similar to their classical counterparts, such properties play an important role in the analysis of the relevant differential operators (both linear and nonlinear). To motivate our results, we recall some classical inequalities. Let E C R be a Caccioppoli set (a measurable set having a locally finite perimeter); then one has the isoperimetric inequality

Complemented Brunn–Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures

Introduction.- Part I Markov semigroups, basics and examples: 1.Markov semigroups.- 2.Model examples.- 3.General setting.- Part II Three model functional inequalities: 4.Poincare inequalities.- 5.Logarithmic Sobolev inequalities.- 6.Sobolev inequalities.- Part III Related functional, isoperimetric and transportation inequalities: 7.Generalized functional inequalities.- 8.Capacity and isoperimetry-type inequalities.- 9.Optimal transportation and functional inequalities.- Part IV Appendices: A.Semigroups of bounded operators on a Banach space.- B.Elements of stochastic calculus.- C.Some basic notions in differential and Riemannian geometry.- Notations and list of symbols.- Bibliography.- Index.

New inequalities for certain Green's functions are given. They may be interpreted physically in many ways, for example, as applying to the quantum mechanical motion of a particle in a potential or to diffusion in the presence of absorbers. These inequalities involve a symmetrization process very closely related to Steiner symmetrization used in the theory of isoperimetric inequalities. The usual geometrical and physical isoperimetric inequalities are very special cases of our general inequality (3.9), arising when the potential is taken to be a characteristic function of a bounded domain and the ``time'' in the Green's function is allowed to get very large or very small.