Groups satisfy linear surface isoperimetric functions.
problem Isoperimetric functions for surface diagrams in hyperbolic groups.
method Analyzing word-hyperbolic groups and their surface diagrams.
result Linear isoperimetric functions for all surface types in hyperbolic groups.
Solves relative isoperimetric problem on polygonal domains, focusing on corners.
problem Relative isoperimetric problem on polygonal domains in R2. method Developed techniques for polygonal domains, with special attention to corners.
result Solved the relative isoperimetric problem for a square with a square corner removed.
The isoperimetric inequality and related inequalities are explored.
problem Proving the isoperimetric inequality and related inequalities.
method Discussing classical and recent proofs.
result Various proofs of the isoperimetric inequality and Sobolev inequality.
Isoperimetric regions in scaled product manifolds are products of regions in each factor.
problem Characterizing isoperimetric regions in anisotropically scaled product manifolds.
method Analyzing regions with smooth boundaries in scaled product manifolds.
result Isoperimetric regions in scaled product manifolds are products of regions in each factor.
The paper finds new inequalities for convex polygons.
problem Finding precise inequalities for convex polygons.
method Analytic isoperimetric inequalities based on Schur convex functions, followed by Bonnesen-style and inverse Bonnesen-style inequalities.
result Sharp discrete isoperimetric inequalities for planar convex polygons.
Huisken's isoperimetric mass is always nonnegative.
problem Understanding the nonnegativity of Huisken's isoperimetric mass.
method Simple reasoning based on basic properties.
result Huisken's isoperimetric mass is nonnegative.
In curved spaces, isoperimetric sets don't exist for small volumes.
problem Nonexistence of isoperimetric sets in spaces of positive curvature.
method Constructing specific noncompact smooth Riemannian manifolds with positive curvature.
result Nonexistence of isoperimetric sets for small volumes in spaces of positive curvature.
Simplified proof for Cheeger's isoperimetric constant.
problem Cheeger's isoperimetric constant
method Simplified proof of Buser's result
result Simplified proof for Cheeger's isoperimetric constant
The study proves the existence and properties of isoperimetric clusters in Riemannian manifolds with bounded geometry.
problem Proving the existence and properties of isoperimetric clusters in Riemannian manifolds with bounded geometry.
method Proved the existence of isoperimetric clusters and compactness theorem for sequence of clusters, introduced Holder continuity of multi-isoperimetric profile.
result Existence and properties of isoperimetric clusters in Riemannian manifolds with bounded geometry.
Study improves isoperimetric inequality for random groups.
problem Improving isoperimetric inequality for random groups.
method Generalizing the inequality to non-planar diagrams.
result Non-planar isoperimetric inequality established.
The paper proves conditions for isoperimetric regions in curved spaces.
problem Finding isoperimetric regions in curved spaces with specific curvature and growth conditions.
method Combining asymptotic mass decomposition, sharp isoperimetric inequality, and concavity property.
result Isoperimetric regions always exist under certain conditions.
The paper proves new inequalities in hyperbolic space using Euclidean methods.
problem Proving weighted isoperimetric inequalities in hyperbolic space.
method Using isoperimetric inequality with log-convex density in Euclidean space.
result Removed horo-convex assumption and proved new inequalities for star-shaped domains.
The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.
problem Finding the minimum perimeter for polygons with a fixed area in hyperbolic geometry.
method Proving analogues of the discrete isoperimetric inequality for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons.
result Established two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with certain area or perimeter restrictions.
We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body C⊂Rn+1, without assuming any further regularity on the boundary of C. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of…
Proves optimal isoperimetric inequality in de Sitter space.
problem Optimal isoperimetric inequality for specific hypersurfaces in de Sitter space.
method Analyzes spacelike, compact, star-shaped, and 2-convex hypersurfaces in de Sitter space.
result Proves an optimal isoperimetric inequality for the specified hypersurfaces.
Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality for Finsler manifolds with specific curvature properties.
method Analyzing measured Finsler manifolds with non-negative Ricci curvature and Euclidean volume growth.
result Sharp isoperimetric inequality and rigidity results for the inequality.
Generic smooth boundaries for isoperimetric regions in 8D manifolds.
problem Understanding boundaries of isoperimetric regions in high-dimensional spaces.
method Generic regularity results for isoperimetric regions in closed Riemannian manifolds of dimension eight.
result Smooth nondegenerate boundaries for isoperimetric regions for generic metrics and volumes.
The paper characterizes hyperbolic manifolds and graphs verifying a specific isoperimetric inequality.
problem Understanding the relationship between hyperbolicity and isoperimetric inequalities in manifolds and graphs.
method Characterization of hyperbolic manifolds and graphs with isoperimetric inequality, using Gromov boundary.
result Having a pole is a necessary condition for verifying the isoperimetric inequality, which can be removed.
Proof of reverse isoperimetric inequality for black holes.
problem Reverse isoperimetric inequality for black holes in Einstein gravity.
method Geometric-analytic approach.
result Reversal of the usual isoperimetric inequality is explained by curved backgrounds governed by Einstein's equations.
The study finds a special isoperimetric inequality for minimal hypersurfaces in spheres.
problem Establishing a special isoperimetric inequality for minimal hypersurfaces in spheres.
method Analyzing the scalar curvature and nodal set of the height function.
result Uniform lower bound for the isoperimetric inequality.
Solves equality case in isoperimetric inequality for non-convex domains.
problem Equality case in relative isoperimetric inequality outside convex sets.
method Analyzes non-convex domains to settle the equality case.
result Solves the equality case for relative isoperimetric inequality outside arbitrary convex sets.
Isoperimetric regions minimize the size of their boundaries among all regions with the same volume. In Euclidean and Hyperbolic space, isoperimetric regions are round balls. We show that isoperimetric regions in two and three-dimensional nonpositively curved manifolds are not necessarily balls, and need not even be con…
Study connects isoperimetric sets, mass concepts, and nonnegative scalar curvature.
problem Understanding mass concepts in nonnegative scalar curvature manifolds.
method Review and analysis of isoperimetric sets and their role in mass concepts.
result Equivalence among mass concepts holds for isoperimetric sets with connected boundaries.
Prove an isoperimetric inequality for compact bodies in 3D contact non-unimodular Lie groups.
problem Isoperimetric inequality for compact bodies in contact non-unimodular 3D Lie groups.
method Prove an isoperimetric inequality.
result Prove an isoperimetric inequality.
Local isoperimetric inequality holds for balls with nonpositive curvature.
problem Preserving the isoperimetric ratio in perturbed ball metrics with nonpositive curvature.
method Analyzing perturbations of ball metrics with nonpositive curvature.
result Isoperimetric ratio is preserved only by homotheties of the ball.
New isoperimetric inequalities in the plane with radial weights identified.
problem Finding isoperimetric shapes in the plane with radial power weights.
method Analyzing the punctured plane with specific volume and perimeter densities.
result Centred balls are uniquely isoperimetric under certain conditions.
New findings on metric spaces with finite Nagata dimension.
problem Understanding isoperimetric properties in subsets of metric spaces.
method Analyzing quasiconvex subsets with finite Nagata dimension and applying isoperimetric inequalities.
result Quasiconvex subsets of metric spaces with finite Nagata dimension are isoperimetrically undistorted up to a certain dimension.
Study on isoperimetric problem in Randers planes achieving maximum area.
problem Isoperimetric problem in Randers planes.
method Analyzing circles centered at the origin for maximum area.
result Circles centered at the origin achieve local maximum area.
We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivit…
The paper finds singular isoperimetric regions in high-dimensional spaces.
problem Existence of singular isoperimetric regions in high-dimensional manifolds.
method Construction of specific manifolds with singular isoperimetric regions.
result Construction of manifolds with singular isoperimetric regions.
We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of t…
We solve the isoperimetric problem in the Lens spaces with large fundamental group. Namely, we prove that the isoperimetric surfaces are geodesic spheres or tori of revolution about geodesics. We also show that the isoperimetric problem in L(3,1) and L(3,2) follows from the proof of the Willmore conjecture by Marques a…
Based on Markvorsen and Palmer's work on mean time exit and isoperimetric inequalities we establish slightly better isoperimetric inequalities and mean time exit estimates for minimal submanifolds of N×R. We also prove isoperimetric inequalities for submanifolds of Hadamard spaces with tamed second fund…
Smooth minimizers found for Willmore energy surfaces.
problem Finding minimizers for Willmore energy surfaces.
method Existence and smoothness established through axially symmetric surfaces with prescribed isoperimetric ratio.
result Existence and smoothness of minimizers proven.
Study minimizers in large volume isoperimetric problems with a new flatness criterion.
problem Minimizers in isoperimetric problems with a compact obstacle.
method Study Plateau-type problem with free boundary, develop mesoscale flatness criterion.
result Identify isoperimetric residue in energy expansion for large volume.
Study proves topological properties of isoperimetric sets in specific spaces.
problem Characterizing isoperimetric sets in PI spaces with deformation property.
method Proves topological regularity results using perimeter increment control.
result Isoperimetric sets are open, have boundary density estimates, and are bounded.
For a complete noncompact connected Riemannian manifold with bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible applications, we extend properties of the isoperimetric profile from compact manifolds to…
Constructs isoperimetric regions from separating hypersurfaces.
problem Finding isoperimetric regions in closed manifolds.
method Constructs isoperimetric regions from separating hypersurfaces.
result Yields isoperimetric boundaries with diverse topological types and singular sets.
Sharp bounds for curve isoperimetric deficit derived.
problem Finding sharp bounds for the isoperimetric deficit of curves.
method Fourier analysis applied to derive Wirtinger-type inequalities.
result Sharp lower and upper bounds for the isoperimetric deficit proved.
Locally isoperimetric partitions minimize perimeter in space.
problem Finding minimal perimeter partitions in space.
method Proving closure theorem to limit sequences of isoperimetric clusters.
result Examples of isoperimetric partitions in various dimensions.
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…
Overview of recent results on isoperimetric inequalities on manifolds with Ricci lower bounds.
problem Isoperimetric problem on manifolds with Ricci lower bounds.
method Modern tools and ideas from nonsmooth geometry.
result Sharp second order differential inequalities for isoperimetric profile.
The article proves Randers Poincaré disc satisfies isoperimetric equality.
problem Extending Riemannian isoperimetric equality to Finslerian case.
method Analyzes Randers Poincaré disc with different volume forms.
result Osserman's result cannot be extended to Finslerian case.
Study finds optimal loops in hyperbolic space with Finsler structure.
problem Optimal loops in Finsler hyperbolic plane.
method Left-invariant Finsler structure, convex trigonometry functions.
result Optimal isoperimetric loops found in terms of trigonometry functions.
The study proves optimal isoperimetric regions in manifolds with density.
problem Finding optimal regions with minimal boundary area in manifolds with density.
method Proving existence of isoperimetric regions and using subgroup actions.
result Isoperimetric regions in product manifolds are slabs.
In this paper we consider the isoperimetric profile of convex cylinders K×Rq, where K is an m-dimensional convex body, and of cylindrically bounded convex sets, i.e, those with a relatively compact orthogonal projection over some hyperplane of Rn+1, asymptotic to a right convex cylind…
Study compares isoperimetric profiles on manifolds with integral Ricci curvature bounds.
problem Comparing isoperimetric profiles on manifolds with integral Ricci curvature bounds.
method Extending previous work, the study uses integral bounds on Ricci curvature to prove comparison results for isoperimetric profile functions.
result Comparison results for the Isoperimetric profile function in manifolds with integral bounds on Ricci curvature.
We give a new proof of the isoperimetric inequality in the plane, based on Steiner's formula for the area of a convex neighborhood. This proof establishes the isoperimetric inequality directly, without requiring that we separately establish the existence of an optimal domain. In doing so, this proof bypasses the main d…