On the isoperimetric deficit in Gauss space (original) (raw)

A quantitative dimension free isoperimetric inequality for the fractional Gaussian perimeter

2020

The Gaussian isoperimetric inequality states that among all sets with prescribed Gaussian measure, the halfspace is the one with least Gaussian perimeter. This result has been proved independently by Borell [6] and Sudakov-Tsirelson [34]. In [12] it has been proved that halfspaces are the only volume-constrained minimizers for the Gaussian perimeter, while in [3,4,13] inequalities of quantitative type, that allow to relate the deficit between a halfspace and a set with the same Gaussian volume with some function of the Gaussian measure of their symmetric difference, are proved. The results in [13] have been improved in [29, 30]. On the other side, fractional perimeters and nonlocal perimeters depending on more general kernels have been object of great attention in the last years, since they are related to nonlocal minimal surfaces [9, 28], phase transitions [35], fractal sets [25]

The Gaussian Isoperimetric Inequality and Transportation

2003

Any probability measure on R d which satisfies the Gaussian or exponential isoperimetric inequality fulfils a transportation inequality for a suitable cost function. Suppose that W (x) dx satisfies the Gaussian isoperimetric inequality: then a probability density function f with respect to W (x) dx has finite entropy, provided that ∇f {log + ∇f } 1/2 W (x) dx < ∞. This strengthens the quadratic logarithmic Sobolev inequality of Gross (Amr. J. Math 97 (1975) 1061). Let µ(dx) = e −ξ(x) dx be a probability measure on R d , where ξ is uniformly convex. Talagrand's technique extends to monotone rearrangements in several dimensions (Talagrand, Geometric Funct. Anal. 6 (1996) 587), yielding a direct proof that µ satisfies a quadratic transportation inequality. The class of probability measures that satisfy a quadratic transportation inequality is stable under multiplication by logarithmically bounded Lipschitz densities.

A strong form of the quantitative isoperimetric inequality

Calculus of Variations and Partial Differential Equations, 2013

We give a refinement of the quantitative isoperimetric inequality. We prove that the isoperimetric gap controls not only the Fraenkel asymmetry but also the oscillation of the boundary.

An isoperimetric inequality for Gauss-like product measures

Journal de Mathématiques Pures et Appliquées, 2016

This paper deals with various questions related to the isoperimetic problem for smooth positive measure dµ = ϕ(x)dx, with x ∈ Ω ⊂ R N. Firstly we find some necessary conditions on the density of the measure ϕ(x) that render the intersection of half spaces with Ω a minimum in the isoperimetric problem. We then identify the unique isoperimetric set for a wide class of factorized finite measures. These results are finally used in order to get sharp inequalities in weighted Sobolev spaces and a comparison result for solutions to boundary value problems for degenerate elliptic equations.

On the isoperimetric problem in Euclidean space with density

Calculus of Variations and Partial Differential Equations, 2007

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp(|x| 2 ) by using symmetrization techniques. I f (V) = inf {P(Ω) : Ω is a smooth open set with vol(Ω) = V}. An isoperimetric region -or simply a minimizer-of volume V is an open set Ω such that vol(Ω) = V and P(Ω) = I f (V).

A Selection Principle for the Sharp Quantitative Isoperimetric Inequality

Archive for Rational Mechanics and Analysis, 2012

We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in R n . Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in R 2 in the small asymmetry regime.

Approximate Gaussian Isoperimetry for k Sets

Lecture Notes in Mathematics, 2012

Given 2 ≤ k ≤ n, the minimal (n − 1)-dimensional Gaussian measure of the union of the boundaries of k disjoint sets of equal Gaussian measure in R n whose union is R n is of order √ log k. A similar results holds also for partitions of the sphere S n−1 into k sets of equal Haar measure.

On isoperimetric sets of radially symmetric measures

Concentration, Functional Inequalities and Isoperimetry, 2011

We study the isoperimetric problem for the radially symmetric measures. Applying the spherical symmetrization procedure and variational arguments we reduce this problem to a one-dimensional ODE of the second order. Solving numerically this ODE we get an empirical description of isoperimetric regions of the planar radially symmetric exponential power laws. We also prove some isoperimetric inequalities for the log-convex measures. It is shown, in particular, that the symmetric balls of large size are isoperimetric sets for strictly log-convex and radially symmetric measures. In addition, we establish some comparison results for general log-convex measures.

Sharp isoperimetric inequalities and sphere theorems

Pacific Journal of Mathematics, 2005

Various relations between sharp isoperimetric inequalities and volumes of manifolds are studied. In particular, we introduce and estimate sharp isoperimetric constants τ * and γ * corresponding to two types of isoperimetric inequalities. We show that for a complete n-dimensional manifold M with Ricci curvature Ric(M) ≥ n -1, the volume of M is close to that of S n if and only if τ * is close to n(n -1)/ 2(n + 2)ω 2/n n and M is simply connected (for n = 2 or 3), or γ * is close to 1 (for any n ≥ 2).

The Isoperimetric Inequality for Compact Rank One Symmetric Spaces and Beyond

arXiv: Metric Geometry, 2017

In this paper, we generalize the spherical Gromov-Milman's isoperimetric inequality to general (closed) Riemannian manifolds. Our isoperimetric inequality also results from needle decomposition and localization methods. This is possible due to Klartag's recent needle decomposition method which works on every closed Riemannian manifold. As a result of our main theorem, we obtain sharp isoperimetric inequalities for compact rank one symmetric spaces (CROSS). Namely, for the complex projective space mathbbCPn\mathbb{C}P^nmathbbCPn and the quaternionic projective space mathbbHPn\mathbb{H}P^nmathbbHPn, we demonstrate that the isoperimetric regions are geodesic balls. For the real projective space mathbbRPn\mathbb{R}P^nmathbbRPn, the isoperimetric regions are given by either the geodesic balls or tubes around some mathbbRPksubsetmathbbRPn\mathbb{R}P^k\subset\mathbb{R}P^nmathbbRPksubsetmathbbRPn.

Concentration of measure and isoperimetric inequalities in product spaces

Publications mathématiques de l'IHÉS, 1995

The concentration of measure prenomenon roughly states that, if a set A in a product Ω N of probability spaces has measure at least one half, "most" of the points of Ω N are "close" to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word "most" is made rigorous by isoperimetrictype inequalities that bound the measure of the exceptional sets. The meaning of the work "close" is defined in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, this simple approach not only yields qualitatively optimal results, but, in many cases, captures near optimal numerical constants. A large number of applications are given, in particular in Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools.