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.
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.
Quantifies fractional isoperimetric inequality with strong control over boundary oscillation.
problem Fractional isoperimetric inequality and its quantitative aspects.
method Regularization process with a new spirit.
result Stability estimates for fractional Cheeger inequality.
In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we…
Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.
problem Understanding isoperimetric constants in metric measure spaces with measure contraction property.
method Proves local isoperimetric inequalities on essentially non-branching MCP(K,N) spaces with volume constraints and geometric conditions.
result Establishes bounds on isoperimetric constants in smaller geodesic balls.
We present a new and direct proof of the local Neumann isoperimetric inequality on convex domains of a Riemannian manifold with Ricci curvature bounded below.
The paper proves pseudolocality theorems for Ricci flows on incomplete manifolds.
problem Pseudolocality of Ricci flows on incomplete manifolds.
method Proves pseudolocality theorems for Ricci flows under specific curvature and isoperimetric conditions.
result Constructs solutions of Ricci flow in balls with pseudolocality property.
The paper proves local rigidity theorems for scalar curvature and related inequalities.
problem Proving local rigidity theorems for scalar curvature and related inequalities.
method Using Ricci flow, the paper studies local rigidity theorems regarding scalar curvature, isoperimetric constant, and logarithmic Sobolev inequality.
result If certain conditions on scalar curvature and isoperimetric constant are met, the metric is locally rigid to Euclidean space.
We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with Ric∞≥1. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle deco…
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.
Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actu…
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.
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 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.
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…
Sharp Sobolev inequality derived for Riemannian manifolds with bounded Ricci curvature.
problem Deriving a sharp Sobolev inequality for Riemannian manifolds with bounded Ricci curvature.
method Reduction to functions with small volume support, first order uniform asymptotic expansion of isoperimetric profile, local uniform Sobolev inequality.
result Sharp Sobolev inequality for W1,p(M) into Ln−pnp(M) is derived. 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…
Proves the Weyl law for 1-cycles in manifolds.
problem Proving the Weyl law for the volume spectrum of 1-cycles in n-dimensional manifolds.
method Using parametric versions of the coarea inequality and isoperimetric inequality, along with a localized approximation method.
result Proves the Weyl law for 1-cycles in manifolds.
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 inequalities on curved spaces without global curvature bounds.
problem Proving inequalities on manifolds with non-negative curvature outside compact sets.
method ABP method localized to regions of non-negative curvature, spectral properties of manifolds.
result Validated isoperimetric and Michael-Simon inequalities on manifolds with asymptotically non-negative curvature.
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 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.
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.
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.
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.
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.
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.
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…
Unified approach to discrete and smooth isoperimetric inequalities of arbitrary order.
problem Finding higher order isoperimetric inequalities for both discrete and smooth curves.
method Unified approach via Fourier analysis of linear operators.
result Unified upper and lower bounds for isoperimetric deficit in smooth curves.
Sharp curvature inequality extends Euclidean isoperimetric inequality to Cartan-Hadamard manifolds.
problem Proving the Euclidean isoperimetric inequality on Cartan-Hadamard manifolds with nullity.
method Using the Chern-Gauss-Bonnet theorem to establish a sharp inequality for total curvature.
result The Euclidean isoperimetric inequality extends to Cartan-Hadamard manifolds with nullity.
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.
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.
The study extends isoperimetric inequalities to non-positive curvature spaces.
problem Isoperimetric inequalities in spaces of non-positive curvature.
method Analyzes submanifolds and geodesics in Cartan-Hadamard manifolds.
result Extensions of isoperimetric inequalities to non-positive curvature spaces.
Proves inequality for maximal spacelike submanifolds in Minkowski space.
problem Maximal spacelike submanifolds in Minkowski space.
method Based on recent work by Brendle.
result Proves isoperimetric-type inequality.
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…
We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) versio…
Sharp inequality outside ball proved using Neumann method.
problem Anisotropic isoperimetric inequality for domains outside an Euclidean ball.
method Applied ABP method to Neumann boundary value problem.
result Proved sharp anisotropic isoperimetric inequality.
Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.
problem Eigenvalue inequalities of Witten-Laplacian on bounded domains.
method Rearrangement technique and trial functions under fixed weighted volume constraint.
result Several isoperimetric inequalities for eigenvalues of Witten-Laplacian.
New inequalities for convex curves with multiple geometric factors.
problem Establishing inequalities for convex curves with multiple geometric factors.
method Parametric isoperimetric-type inequalities for closed convex curves with parameter conditions and equality conditions.
result Derived new inequalities and improved versions of existing inequalities.
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.
problem Finding bounds for eigenvalues of Neumann Laplacian on domains in symmetric spaces.
method Proving sharp inequalities for eigenvalues in compact and noncompact rank-1 symmetric spaces.
result Generalization of previous results for hyperbolic space and symmetric spaces.
Sharp stability results for reverse isoperimetric inequalities in 2D.
problem Reverse isoperimetric inequalities in the plane.
method Stability analysis of λ-convex bodies and convex bodies with smooth boundaries. result Sharp stability results for reverse isoperimetric inequalities, including inradius and Cheeger inequalities.
Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality on Finsler metric measure manifolds.
method Defining volume entropy and second Cheeger constant, proving sharp inequality.
result Sharp isoperimetric inequality involving volume entropy and weighted Ricci curvature.
New method uses entropy dissipation to prove isoperimetric inequalities.
problem Proving isoperimetric inequalities in geometric settings.
method Information-theoretic approach based on entropy dissipation under heat flow.
result New proof of Euclidean isoperimetric inequality with sharp constant.
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.
We develop the celebrated semigroup approach à la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenböck formula established by Sturm and the author. We show the L1-gradient estimate on Finsler manifolds (under some additional assumptions in the n…