Related papers: On The Quantitative Isoperimetric Inequality In Th…

We prove a sharp quantitative version of the Faber--Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit $\delta(f;\Omega)$ which measures by how much the STFT of a function $f\in L^2(\mathbb R)$…

The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp ?weighted isoperimetric inequality? and a nonsharp classical inequality similar to the…

We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in $R^n$. Combining these results with a refinement of the selection principle introduced by the authors in arXiv:0911.0786, we describe a…

The Bergman $p$-analytic content ($1\leq p<\infty $) of a planar domain $\Omega $ measures the $L^{p}(\Omega )$-distance between $\overline{z}$ and the Bergman space $A^{p}(\Omega )$ of holomorphic functions. It has a natural analogue in…

We prove an isoperimetric inequality for the second non-zero eigenvalue of the Laplace-Beltrami operator on the real projective plane. For a metric of the unit area this eigenvalue is not greater than 20\pi. This value is attained in the…

We show a reverse isoperimetric inequality within the class of relative outer parallel bodies, with respect to a general convex body $E$, along with its equality condition. Based on the convexity of the sequence of quermassintegrals of…

In this article, we investigate the centered isoperimetric inequality on Cartan-Hadamard manifolds endowed with a warped product structure, namely, among all bounded measurable sets of finite perimeter and prescribed volume, the geodesic…

We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded domain $\Omega \subset \mathbb R^n$. More precisely, we prove that if the fractional torsion…

We show a strong version of the fractional quantitative isoperimetric inequality, in which the isoperimetric deficit controls not only the Fraenkel asymmetry but also a sort of oscillation of the boundary. This generalizes the local result…

Let $\Omega$ be a smooth, bounded domain in $\mathbb R^3$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $\Omega$ is at most the Yamabe quotient of a ball, and equality holds if and only if $\Omega$ is…

An isoperimetric inequality on the Hamming cube for exponents $\beta\ge 0.50057$ is proved, achieving equality on any subcube. This was previously known for $\beta\ge \log_2(3/2)\approx 0.585$. Improved bounds are also obtained at the…

Over the past few decades, experimental tests of Bell-type inequalities have been at the forefront of understanding quantum mechanics and its implications. These strong bounds on specific measurements on a physical system originate from…

We present a new proof of the Willmore inequality for an arbitrary bounded domain $\Omega\subset\mathbb{R}^{n}$ with smooth boundary. Our proof is based on a parametric geometric inequality involving the electrostatic potential for the…

This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $ \theta ,$ where $ \theta >2$ is real and non-integral. For fixed non-zero real numbers $ \lambda_i $ not all of the same sign we write…

Let $\Omega\subset \mathbb{R}^d$ be an open set of finite measure and let $\Theta$ be a disjoint union of two balls of half measure. We study the stability of the full Dirichlet spectrum of $\Omega$ when its second eigenvalue is close to…

In this paper, the concept of weakly uniform perfectness is considered. As an analogue of the theory of uniform perfectness, we obtain the relationships between weakly uniform perfectness and Bergman kernel, Poincar\'e metric and Hausdorff…

The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the…

We prove that the isoperimetric profile of a convex domain $\Omega$ with compact closure in a Riemannian manifold $(M^{n+1},g)$ satisfies a second order differential inequality which only depends on the dimension of the manifold and on a…

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santalo inequality and the affine isoperimetric inequality…

Let $\Omega$ be a smooth, bounded subset of $\mathbb{R}^3$ diffeomorphic to a ball. Consider $M = \mathbb{R}^3 \setminus \Omega$ equipped with an asymptotically flat metric $g = f^4 g_{\text{euc}}$, where $f\to 1$ at infinity. Assume that…