arXiv research

The paper characterizes hyperbolic manifolds and graphs verifying a specific isoperimetric inequality.

The paper proves conditions for isoperimetric regions in curved spaces.

New inequalities for convex curves with multiple geometric factors.

Study nonnegatively curved Alexandrov spaces, proving isoperimetric conditions and structure at infinity.

In this note, we show that there is some counterexample for isoperimetric inequality if the condition $(φ')^2-φ"φ\leq 1$ does not hold in warped product space.

New isoperimetric inequalities in the plane with radial weights identified.

We give simple conditions on an ambient manifold that are necessary and sufficient for isoperimetric inequalities (for submanifolds) to hold.

In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its e…

Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.

Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.

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…

We use a locally constrained mean curvature flow to prove the isoperimetric inequality for spacelike domains in generalized Robertson-Walker spaces satisfying the null convergence condition.

The paper studies isoperimetric inequalities on warped product manifolds.

We prove existence of isoperimetric regions for every volume in non-compact Riemannian $n$-manifolds $(M,g)$, $n\geq 2$, having Ricci curvature $Ric_g\geq (n-1) k_0 g$ and being locally asymptotic to the simply connected space form of constant sectional curvature $k_0$; moreover in case $k_0=0$ we show that the isoperi…

New method uses entropy dissipation to prove isoperimetric inequalities.

Proves existence of proper solutions for inverse mean curvature flow.

Study the isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to…

The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic geometry as well. In this paper, we generalize the discrete isoperimetric inequality t…

Let $M$ be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of $M$ is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatur…

We prove that the results regarding the Isoperimetric inequality and Cheeger constant formulated in terms of the Minkowski content, obtained by the authors in previous papers in the framework of essentially non-branching metric measure spaces verifying the local curvature dimension condition, also hold in the stronger …

The study solves the isoperimetric problem for Heisenberg group norms.

Adapts Stein's method for geometric inequalities, addressing boundary terms.

Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by opti…

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a…

We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoper…

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 $\mathbb{R} \times_{\cosh(\sqrt{K/(1-N)}t)} Σ^{n-1}$ of hyperbolic nature, where $Σ^{n-1}$ i…

The paper extends inequalities to closed Riemannian manifolds.

New isoperimetric inequality for clamped plates in RCD(0,N) spaces, sharp and stable.

Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.

The aim of this article is: (a) To establish the existence of the best isoperimetric constants for the $(H^1,BMO)$-normal conformal metrics $e^{2u}|dx|^2$ on $\mathbb R^n$, $n\ge 3$, i.e., the conformal metrics with the Q-curvature orientated conditions $$ (-Δ)^{n/2}u\in H^1(\mathbb R^n) & \ u(x)=\hbox{const.}+\frac{\i…

Groups satisfy linear surface isoperimetric functions.

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are shar…

Solves relative isoperimetric problem on polygonal domains, focusing on corners.

The isoperimetric inequality and related inequalities are explored.

The paper investigates higher dimensional analogues of Burago's inequality bounding the area of a closed surface by its total curvature. We obtain sufficient conditions for hypersurfaces in 4-space that involve the Ricci curvature. We get semi-local variants of the inequality holding in any dimension that involve domai…

Isoperimetric regions in scaled product manifolds are products of regions in each factor.

A well known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions. In dimension 2, the constant can be controlled by the integral of the positive part of the Gaussian curvature. In this paper, we showed that on simply connected conformal flat manifol…

The paper finds new inequalities for convex polygons.

Huisken's isoperimetric mass is always nonnegative.

Survey on Allen-Cahn equations and systems, focusing on multiplicity results and geometric interpretation.

In curved spaces, isoperimetric sets don't exist for small volumes.

The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established -- along with its equality conditions.

Simplified proof for Cheeger's isoperimetric constant.

The study proves the existence and properties of isoperimetric clusters in Riemannian manifolds with bounded geometry.

Study improves isoperimetric inequality for random groups.

The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.

The paper proves new inequalities in hyperbolic space using Euclidean methods.