Related papers: A weighted relative isoperimetric inequality in co…

A $\lambda$-convex body in a three-dimensional space form $M^3(c)$ of constant curvature $c$ is a compact convex set $K$ whose boundary $\partial K$ has normal curvatures bounded below by a constant $\lambda>0$ (in a weak sense). Within…

In this paper, Monge-Kantorovich problem is considered in the infinite dimension on an abstract Wiener space $(W, H,\mu)$, where $H$ is Cameron-Martin space and $\mu$ is the Gaussian measure. We study the regularity of optimal transport…

We study the Monge and Kantorovich transportation problems on $\mathbb{R}^{\infty}$ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal…

We consider the monomial weight $x^{A}=\vert x_{1}\vert^{a_{1}}\ldots\vert x_{N}\vert^{a_{N}}$, where $a_{i}$ is a nonnegative real number for each $i\in\{1,\ldots,N\}$, and we establish the existence and nonexistence of isoperimetric…

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…

In this paper, we consider the problem of minimizing a difference-of-convex objective over a nonlinear conic constraint, where the cone is closed, convex, pointed and has a nonempty interior. We assume that the support function of a compact…

We prove, using optimal transport tools, weighted Poincar'e inequalities for log-concave random vectors satisfying some centering conditions. We recover by this way similar results by Klartag and Barthe-Cordero-Erausquin for log-concave…

This paper deals with various questions related to the isoperimetic problem for smooth positive measure $d\mu = \varphi(x)dx$, with $x \in \Omega \subset \mathbb{R}^N$. Firstly we find some necessary conditions on the density of the measure…

We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This…

We study the asymptotics of complete Kaehler-Einstein metrics on strictly pseudoconvex domains in C^n and derive a convergence theorem for solutions to the corresponding Monge-Ampere equation. If only a portion of the boundary is analytic,…

We study the solvability and uniqueness for several degenerate Monge--Amp\`ere equations including the Monge--Amp\`ere eigenvalue problem in real Euclidean spaces that involve singular Borel measures. Our approach systematically analyzes…

In this paper we provide a Bonnesen-style inequality which gives a lower bound for the isoperimetric deficit corresponding to a closed convex curve in terms of some geometrical invariants of this curve. Moreover we give a geometrical…

The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{\'e} constant. First we revisit E. Milman's result [40] on the link between weak (Poincar{\'e} or…

In this note we present the solution of some isoperimetric problems in open convex cones of $\R^n$ in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin…

We consider the punctured plane with volume density $|x|^\alpha$ and perimeter density $|x|^\beta$. We show that centred balls are uniquely isoperimetric for indices $(\alpha,\beta)$ which satisfy the conditions $\alpha-\beta+1>0$,…

The sharp isoperimetric inequality for non-compact Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions…

Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to…

While the shape equations describing the equilibrium of an unstretchable thin sheet that is free to bend are known, the boundary conditions that supplement these equations on free edges have remained elusive. Intuitively, unstretchability…

In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $\Omega$, different from a ball, which minimizes the ratio $\delta(\Omega)/\lambda^2(\Omega)$, where $\delta$ is the…

We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex…