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.
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 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.
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 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.
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.
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.
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.
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.
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.
The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdré). Then we give a quan…
Optimally estimates stability in Lorentzian isoperimetric inequalities.
problem Stability estimates in Lorentzian isoperimetric inequalities.
method Quantitative stability estimates using Fraenkel asymmetry and Lipschitz bounds.
result Optimal stability estimates with universal constants for Lorentzian isoperimetric inequalities.
Paper proves a new isoperimetric inequality for Steklov eigenvalues.
problem Finding a new isoperimetric inequality for Steklov eigenvalues.
method Proving a Brock-type inequality under specific conditions.
result Extension of Brock's classical result to Witten-Laplacian.
The paper studies isoperimetric inequalities on warped product manifolds.
problem Investigating isoperimetric inequalities on warped product manifolds.
method Exploiting the interplay between isoperimetric inequalities and warped product structures, deriving necessary and sufficient conditions.
result Established a quantitative lower bound for the first nonzero Dirichlet eigenvalue of geodesic balls centered at the pole.
Paper finds inequalities for convex domains in hyperbolic space.
problem Finding inequalities for convex domains in hyperbolic space.
method Introducing hyperbolic ellipsoids and using orthogonal projection to establish inequalities.
result Affine isoperimetric inequalities for static convex domains in hyperbolic space characterized by hyperbolic ellipsoids.
Linear inequality found for certain curved spaces.
problem Finding isoperimetric inequalities for curved spaces.
method Extending known result to homogeneous Hadamard manifolds.
result Linear isoperimetric inequality proven for higher-dimensional cycles.
Sharp inequalities in nonnegative Ricci curvature spaces using mass transport.
problem Proving sharp isoperimetric and Sobolev inequalities in nonnegative Ricci curvature spaces.
method Optimal mass transport theory, symmetrization techniques, and volume non-collapsing properties.
result Sharp isoperimetric and Sobolev inequalities established in Riemannian manifolds with nonnegative Ricci curvature.
New proof of Wulff-Gage inequality with applications.
problem Proving the Wulff-Gage isoperimetric inequality.
method Provided a new proof of the inequality for origin-symmetric convex bodies.
result Uniqueness of log-Minkowski problem and new proof of log-Minkowski inequality.
The paper proves new inequalities on the unit ball in higher dimensions.
problem Establishing new weighted inequalities on the unit ball.
method Limiting approach to prove Carleman and Huber inequalities.
result Sharp weighted Carleman and Huber inequalities on the unit ball.
In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature, and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar cu…
Study proves inequality for eigenvalues in symmetric spaces.
problem Proving an inequality for Steklov eigenvalues in symmetric spaces.
method Analyzes eigenvalues on bounded domains in noncompact rank-1 symmetric spaces.
result Extends previous results to non-Euclidean spaces.
In this paper, we prove a Sobolev and isoperimetric inequalities for submanifold in weighted manifold. Our results generalize the Hoffman-Spruck's inequalities.
Proves inequality for Steklov eigenvalues in hyperbolic space.
problem Finding bounds for Steklov eigenvalues in hyperbolic geometry.
method Proves isoperimetric inequality for harmonic mean of eigenvalues.
result Establishes inequality for hyperbolic Steklov eigenvalues.
Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann--Petty centroid inequality.
Quantifies nearly spherical subsets in complex ball geometry.
problem Isoperimetric inequality for nearly spherical domains in Bergman ball.
method Proves a quantitative isoperimetric inequality for nearly spherical subsets of Bergman ball.
result First result on isoperimetric phenomenon in Bergman ball.