Quantitative Isoperimetric Inequality Research Articles

Quantitative isoperimetric inequalities in $$\mathbb {H}^n$$ H n

In the Heisenberg group H^n, we prove quantitative isoperimetric inequalities for Pansu's spheres, that are known to be isoperimetric under various assumptions. The inequalities are shown for suitably restricted classes of competing sets and the proof relies on the construction of sub-calibrations.

A sharp quantitative isoperimetric inequality in higher codimension

We establish a quantitative isoperimetric inequality in higher codimension. In particular, we prove that for any closed (n-1) -dimensional manifold \Gamma in \mathbb R^{n+k} the following inequality \mathbf D(\Gamma)\ge C \mathbf d^2(\Gamma) holds true. Here, \mathbf D(\Gamma) stands for the isoperimetric gap of \Gamma , i.e. the deviation in measure of \Gamma from being a round sphere and \mathbf d(\Gamma ) denotes a natural generalization of the Fraenkel asymmetry index of \Gamma to higher codimensions.

Nonlocal quantitative isoperimetric inequalities

We show a quantitative-type isoperimetric inequality for fractional perimeters where the deficit of the $$t$$ -perimeter, up to multiplicative constants, controls from above that of the $$s$$ -perimeter, with $$s$$ smaller than $$t$$ . To do this we consider a problem of independent interest: we characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $$t$$ -perimeter and the $$s$$ -perimeter. In particular, we show that balls are the unique minimizers if the volume is sufficiently small, depending on $$t-s$$ , while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all $$s,\,t$$ . When $$s=0$$ this problem reduces to the fractional isoperimetric problem, for which it is well known that balls are the only minimizers.

An analytic proof of the planar quantitative isoperimetric inequality

We give an analytic proof of the quantitative isoperimetric inequality in the plane and give an estimation of the upper bound of the constant via maximizing the L∞-norm of the gradient of solutions to the Poisson equation.

Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies

We obtain a sharp quantitative isoperimetric inequality for nonlocal $s$-perimeters, uniform with respect to $s$ bounded away from $0$. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal $s$-perimeter plus a non-local repulsive interaction term. In the particular case $s =1$ the $s$-perimeter coincides with the classical perimeter, and our results improve the ones of Kn\"upfer and Muratov concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term. More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts.

Quantitative isoperimetric inequalities for log-convex probability measures on the line

The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdré). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetric difference between sets) from any extremal sets! From the results on sets we derive quantitative functional inequalities of weak Cheeger type.

A strong form of the quantitative isoperimetric inequality

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.

Minimality via Second Variation for a Nonlocal Isoperimetric Problem

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are $L^1$-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.

Bonnesenʼs inequality for John domains in [formula omitted

We prove sharp quantitative isoperimetric inequalities for John domains in Rn. We show that the Bonnesen-style inequalities hold true in Rn under the John domain assumption which rules out cusps. Our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit. We use the sharp quantitative inequalities proved in Fusco et al. (2008) [7] and Fuglede (1989) [4] to reduce our problem to symmetric domains.

A Selection Principle for the Sharp Quantitative Isoperimetric Inequality

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

Quantitative isoperimetric inequalities and homeomorphisms with finite distortion

We prove quantitative isoperimetric inequalities for images of the unit ball under homeomorphisms of exponentially integrable distortion. We show that the metric distortions of such domains can be controlled by their Fraenkel asymmetries. An application of the quantitative isoperimetric inequality proved by Hall and Fusco, Maggi, and Pratelli then shows that for these domains a version of Bonnesen’s inequality holds in all dimensions.

Un nuovo approccio alle disuguaglianze isoperimetriche quantitative

We introduce a new variational method for studying geometric and functional inequalities with quantitative terms. In the context of isoperimetric-type inequalities, this method (called Selection Principle) is based on a penalization technique combined with the regularity theory of quasiminimizers of the perimeter functional. In this seminar we present the method and describe two remarkable applications. The rst one is a new proof of the sharp quantitative isoperimetric inequality in Rn. The second one is the proof of a conjecture posed by Hall about the optimal constant in the quantitative isoperimetric inequality in R2, in the small asymmetry regime.

Quantitative Isoperimetric Inequalities on the Real Line

In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space. Using only geometric tools, we extend their result to all symmetric log-concave measures \mu on the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter), the intervals or complements of intervals have minimal perimeter.

A mass transportation approach to quantitative isoperimetric inequalities

A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric inequality and the Brenier-McCann Theorem. A sharp quantitative version of the Brunn-Minkowski inequality for convex sets is proved as a corollary.

Sharp quantitative isoperimetric inequalities in the 𝐿¹ Minkowski plane

An isoperimetric inequality bounds from below the perimeter of a domain in terms of its area. A quantitative isoperimetric inequality is a stability result: it bounds from above the distance to an isoperimetric minimizer in terms of the isoperimetric deficit. In other words, it measures how close to a minimizer an almost optimal set must be. The euclidean quantitative isoperimetric inequality has been thoroughly studied, in particular by Hall and by Fusco, Maggi and Pratelli, but the L 1 L^1 case has drawn much less attention. In this note we prove two quantitative isoperimetric inequalities in the L 1 L^1 Minkowski plane with sharp constants and determine the extremal domains for one of them. It is usually (but not here) difficult to determine the extremal domains for a quantitative isoperimetric inequality: the only such known result is for the euclidean plane, due to Alvino, Ferone and Nitsch.

Quantitative isoperimetric inequalities for a class of nonconvex sets

Quantitative versions (i.e., taking into account a suitable “distance” of a set from being a sphere) of the isoperimetric inequality are obtained, in the spirit of Fuglede (Trans Am Math Soc 314:619–638, 1989) and Fusco et al. (Ann Math 168:941–980, 2008) for a class of not necessarily convex sets called φ-convex sets. Our work is based on geometrical results on φ-convex sets, obtained using methods of both nonsmooth analysis and geometric measure theory.

A note on Cheeger sets

Starting from the quantitative isoperimetric inequality, we prove a sharp quantitative version of the Cheeger inequality.

The local homeomorphism property of spatial quasiregular mappings with distortion close to one

We give a quantitative proof for a theorem of Martio, Rickman and Vaisala [MRV] on the rigidity of the local homeomorphism property of spatial quasiregular mappings with distortion close to one. The proof is based on a distortion theory established by using two main tools. First, we use a conformal invariant between sphere families and components of their preimages under quasiregular mappings. Secondly, we use Hall’s quantitative isoperimetric inequality result [H] to relate two different types of distortion.

Confinement of Brownian motion among Poissonian obstacles in ℝ d , d≥ 3

Consider Brownian motion among random obstacles obtained by translating a fixed compact nonpolar subset of ℝ d , d≥ 1, at the points of a Poisson cloud of constant intensity v <: 0. Assume that Brownian motion is absorbed instantaneously upon entering the obstacle set. In SZN-conf Sznitman has shown that in d = 2, conditionally on the event that the process does not enter the obstacle set up to time t, the probability that Brownian motion remains within distance ∼t 1/4 from its starting point is going to 1 as t goes to infinity. We show that the same result holds true for d≥ 3, with t 1/4 replaced by t 1/( d +2). The proof is based on Sznitmans refined method of enlargement of obstacles [10] as well as on a quantitative isoperimetric inequality due to Hall [4].

A quantitative isoperimetric inequality in n-dimensional space.

A quantitative isoperimetric inequality in n-dimensional space.