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.
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.
The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.
problem Analyzing the negative gradient flow of the Willmore energy plus volume.
method Proved a quantitative reverse isoperimetric inequality and applied it to the flow.
result Initial surfaces converge to a round point in finite or infinite time.
The paper solves reverse isoperimetric problems for convex bodies with curvature constraints.
problem Finding the smallest volume among λ-convex bodies of a given surface area. method Using λ-convex bodies and analyzing their properties in model spaces of constant curvature. result The λ-convex lens is the unique minimizer of volume among all λ-convex bodies of given surface area in R3. We consider the class of λ-concave bodies in Rn+1; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius 1/λ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the co…
We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature ⩾c. We also study the equality case.
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…
Sharp inequalities for curved surfaces and cones.
problem Optimizing areas in nonpositively curved spaces.
method Proving inequalities for disks and triangles in cones.
result Minimal area properties for specific shapes in cones.
In 3D space forms, a lens minimizes volume for a fixed surface area.
problem Finding the shape with minimal volume for a given surface area in 3D space forms.
method Proving a sharp reverse isoperimetric inequality for λ-convex bodies. result The λ-convex lens minimizes volume for a fixed surface area in 3D space forms. 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.
We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.
The study proves properties of optimizers for sets maximizing perimeter under fixed volume constraints.
problem Existence and properties of bounded convex sets in Riemannian manifolds maximizing perimeter under fixed volume constraints.
method Analyzes the properties of optimizers for sets maximizing perimeter under fixed volume constraints in Euclidean, spherical, and hyperbolic spaces.
result Proves that there are no C2-maximisers of perimeter with prescribed volume and that the smallest principal curvature is constant in regions where the set is of class C2. 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…
The paper proves reverse inequalities in various geometric settings using curvature radius data.
problem Proving reverse Alexandrov-Fenchel inequalities in different geometric settings.
method Using curvature radius data and associated evolute or focal maps.
result Sharp reverse Alexandrov-Fenchel estimates and inequalities in smooth convex curves and hypersurfaces.
Sharp inequalities for star bodies in 2D space.
problem Understanding star bodies in 2D space.
method Sharp inequalities for star bodies in R2. result New inequalities and proofs for star bodies.
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…
The paper finds the unique minimizer of area for hyperbolic bodies with curvature constraints.
problem Finding the unique minimizer of area for hyperbolic bodies with curvature constraints.
method Introduced the concept of 'thick λ-sausage' bodies and used extra assumption of thickness to handle non-convex inner parallel bodies. result The thick λ-sausage body is the unique minimizer of area among all bodies with a given length and curvature constraints. 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 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 strong volume estimates for convex bodies from data about their sections or projec…
In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.
Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide t…
We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound l for the cut distance and Ricci curvature lower bound (n−1)k, the geodesic ball of radius l in the space form o…
Based on the isoperimetric inequality, G. Huisken proposed a definition of total mass in general relativity that is equivalent to the ADM mass for (smooth) asymptotically flat 3-manifolds of nonnegative scalar curvature, but that is well-defined in greater generality. In a similar vein, we use the isocapacitary inequal…
Sharp Gaussian isoperimetry proven along Ricci flow.
problem Proving sharp Gaussian isoperimetric inequality for Ricci flow.
method Using monotonicity formula to prove inequality.
result Exact Gaussian enlargement theorem and concentration estimates.
We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable…
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.
We prove that the length of the boundary of a J-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the rea…
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.
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.
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.