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).