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, GromovMilman, 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, KannanLovász-Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semigroup 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 BakryEmery.

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 ∞, 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.When q = ∞, the last term in (1.1) is interpreted as 0, whereas when q = 0, this term only makes sense if Ψ is constant, in which case Ric g,Ψ,0 := Ric g . Here as usual Ric g denotes the Ricci curvature tensor and ∇ g denotes the Levi-Civita covariant derivative.1 Definition (Curvature-Dimension-Diameter Condition). (M n , g, µ) is said to satisfy the Curvature-Dimension-Diameter Condition, if µ is supported on the closure of a geodesically convex domain Ω ⊂ M of diameter at most D, having (possibly empty) C 2 boundary, µ = Ψ · vol g | Ω with Ψ > 0 on Ω and log(Ψ) ∈ C 2 (Ω), and as 2-tensor fields:Ric g,Ψ,q ≥ ρg on Ω .When Ω = M and D = +∞, the latter definition coincides with the celebrated Bakry-Emery Curvature-Dimension condition CD(ρ, n + q), introduced in an equivalent form in [3] (in the more abstract framework of diffusion generators). Indeed, the generalized Ricci tensor incorporates information on curvature and dimension from both the geometry of (M, g) and the measure µ, and so ρ may be thought of as a generalized-curvature lower bound, and n + q as a generalized-dimension upper bound. The generalized Ricci tensor (1.1) was introduced with q = ∞ in [50,51] and in general in [2] (the equivalent form (1.2) was noted in [52]), and has been extensively studied and used in recent years (see e.g. also [67,45,73,65,7,70,53,76,62] and the references therein).In this work, we obtain a sharp isoperimetric inequality on (M n , g, µ) under the CDD(ρ, n+ q, D) condition, for the entire range of parameters ρ ∈ R, q ∈ [0, ∞], D ∈ (0, ∞], in a single unified framework. In particular, for each choice of parameters, we identify the model spaces which are extremal for the isoperimetric problem. Our results seem new even in the classical constant-density case (q = 0) when ρ ≤ 0 and D < ∞ or when ρ > 0 and D < π (n − 1)/ρ. We start by recalling the notion of an isoperimetric inequality in a general measure-metric space settin...

The Lott-Sturm-Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space (X, d, m) (so that (supp(m), d) is a length-space and m(X) < ∞) verifying the local Curvature-Dimension condition CD loc (K, N ) with parameters K ∈ R and N ∈ (1, ∞), also verifies the global Curvature-Dimension condition CD(K, N ), meaning that the Curvature-Dimension condition enjoys the globalization (or local-to-global) property. The main new ingredients of our proof are an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order derivative of an associated Kantorovich potential; a surprising third-order bound on the latter Kantorovich potential, which holds in complete generality on any proper geodesic space; and a certain rigidity property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. The change-of-variables formula is obtained via a new synthetic notion of Curvature-Dimension we dub CD 1 (K, N ). Contents

Given an isotropic random vector X with log-concave density in Euclidean space R n , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular thatfor some universal constants c, C > 0. This improves the best known deviation results on the thin-shell and mesoscopic scales due to Fleury and Klartag, respectively, and recovers the sharp large-deviation estimate of Paouris. Another new feature of our estimate is that it improves when X is ψ α (α ∈ (1, 2]), in precise agreement with Paouris' estimates. The upper bound on the thin-shell width Var(|X|) we obtain is of the order of n 1/3 , and improves down to n 1/4 when X is ψ 2 . Our estimates thus continuously interpolate between a new best known thin-shell estimate and the sharp large-deviation estimate of Paouris. As a consequence, a new best known bound on the Cheeger isoperimetric constant appearing in a conjecture of Kannan-Lovász-Simonovits is deduced.

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. 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 best possible (up to constants), and significantly extend all previously known results, which could only deduce dimension dependent bounds, or could not deduce anything stronger than a linear isoperimetric inequality in the restrictive non-negative 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.

We study the isoperimetric, functional and concentration properties of n-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension N is negative, and more generally, is in the range N ∈ (−∞, 1), extending the scope from the traditional range N ∈ [n, ∞]. 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 -sphere and Gaussian measure when N ∈ [n, ∞]): a (positively curved) sphere of (possibly negative) dimension N ∈ (−∞, 1). When curvature is non-negative, we show that arbitrarily weak concentration implies an 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], and enjoy a two-level concentration of the type 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). IntroductionLet (M n , g) denote an n-dimensional (n ≥ 2) complete connected oriented smooth Riemannian manifold with (possibly empty) boundary, and let µ denote a measure on M having density Ψ with respect to the Riemannian volume form vol g . We assume that M is geodesically convex (any two points may be connected by a distance minimizing geodesic), that ∂M is C 2 smooth, and that Ψ is positive and C 2 smooth on the entire M (all the way up to the boundary). As usual, we denote by Ric g the Ricci curvature tensor and by ∇ g the Levi-Civita covariant derivative.

It is known that by dualizing the Bochner-Lichnerowicz-Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Brascamp-Lieb type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp-Lieb, Bobkov-Ledoux and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Our framework allows to encompass the entire class of Borell's convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative generalized dimension.

We unify and slightly improve several bounds on the isotropic constant of highdimensional convex bodies; in particular, a linear dependence on the body's ψ 2 constant is obtained. Along the way, we present some new bounds on the volume of L p -centroid bodies and yet another equivalent formulation of Bourgain's hyperplane conjecture. Our method is a combination of the L p -centroid body technique of Paouris and the logarithmic Laplace transform technique of the first named author.