Related papers: A weighted isoperimetric inequality in a wedge

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…

Given $2\le k\le n$, 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 $\sqrt{\log k}$. A similar results holds also…

Let $\mu$ be a probability measure on $\mathbb{R}^n$ with a bounded density $f$. We prove that the marginals of $f$ on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there…

Early results by Borel and Cantelli and Erd\H{o}s and Chung have provided bounds for the measure of a limsup set in terms of measures of its constituent sets and their intersections. Recent work by Beresnevich and Velani \cite{Velanipaper}…

We consider the Gaussian correlation inequality for nonsymmetric convex sets. More precisely, if $A\subset\mathbb{R}^d$ is convex and the origin $0\in A$, then for any ball $B$ centered at the origin, it holds $\gamma_d(A\cap B)\geq…

Let $G = (V,E)$ be a connected graph. A probability measure $\mu$ on $V$ is called "balanced" if it has the following property: if $T_\mu(v)$ denotes the "earth mover's" cost of transporting all the mass of $\mu$ from all over the graph to…

Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $\nu_n = \sum_i \xi_i $, where $\xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near…

In a metric space $(X,d)$ we reconstruct an approximation of a Borel measure $\mu$ starting from a premeasure $q$ defined on the collection of closed balls, and such that $q$ approximates the values of $\mu$ on these balls. More precisely,…

Upper bounds are obtained for the Newtonian capacity of compact sets in $\R^d,\,d\ge 3$ in terms of the perimeter of the $r$-parallel neighbourhood of $K$. For compact, convex sets in $\R^d,\,d\ge 3$ with a $C^2$ boundary the Newtonian…

We study, on a weighted Riemannian manifold of Ric$_{N} \geq K > 0$ for $N < -1$, when equality holds in the isoperimetric inequality. Our main theorem asserts that such a manifold is necessarily isometric to the warped product $\mathbb{R}…

We study a function, which is a weighted sum of the squares of the distances of an arbitrary point to the sidelines of a triangle. The given weights, considered as barycentric coordinates, determine a point $M$. We prove that the function…

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of…

We prove that on an essentially non-branching $\mathrm{MCP}(K,N)$ space, if a geodesic ball has a volume lower bound and satisfies some additional geometric conditions, then in a smaller geodesic ball (in a quantified sense) we have an…

Using recent results of Berman and Boucksom we show that for a non-pluripolar compact set K in C^d and an admissible weight function w=e^{-\phi} any sequence of so-called optimal measures converges weak-* to the equilibrium measure…

In this paper we investigate the reverse isoperimetric inequality with respect to the Gaussian measure for convex sets in $\mathbb{R}^{2}$. While the isoperimetric problem for the Gaussian measure is well understood, many relevant aspects…

We consider the isoperimetric inequality involving the $s$-perimeter and the $t$-perimeter with $0<s<t<1$, and show that the ball is a local minimizer of the (scale-invariant) isoperimetric ratio $\mathcal{F}(E):=P_t(E)^{\frac{1}{n-t}}/…

In this note we study the maximal perimeter of a convex set in $\mathbb{R}^n$ with respect to various classes of measures. Firstly, we show that for a probability measure $\mu$ on $ \mathbb{R}^n$, satisfying very mild assumptions, there…

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on…

We give a solution to the isoperimetric problem for the exponential measure on the plane with the $\ell_1$-metric. As it turns out, among all sets of a given measure, the simplex or its complement (i.e. the ball in the $\ell_1$-metric or…

Let $(X,\mathcal{A}, \mu)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be invertible bi measurable measure preserving transformations on this measure space. We give a sufficient condition for the product of $H$ bounded…