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.
Sharp isoperimetric inequality proven for specific metric measure spaces.
problem Proving isoperimetric inequality in metric measure spaces with synthetic conditions.
method Synthetic condition called Measure-Contraction property; Lévy-Gromov inequality.
result Sharp isoperimetric inequality holds true for spaces with synthetic conditions.
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.
The study solves the isoperimetric problem for Heisenberg group norms.
problem Solving the isoperimetric problem for anisotropic norms in the Heisenberg group.
method Representation formula for perimeter, foliation property, differential equation characterization, approximation procedure.
result Characterization of isoperimetric sets as sub-Finsler analogues of Pansu's bubbles.
Paper improves isoperimetric inequality with new curve properties.
problem Classical isoperimetric inequality in Euclidean plane.
method Introduced improved isoperimetric inequality for convex curves.
result Improved inequality involving Wigner caustic and oriented areas.
Metric spaces with non-positive curvature have a specific curve isoperimetric property.
problem Understanding spaces with non-positive curvature.
method Proving isoperimetric inequality for metric spaces with Alexandrov curvature.
result Metric spaces with non-positive curvature have a specific curve isoperimetric property.
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…
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 paper develops new methods to study sharp isoperimetric properties on complex spaces.
problem Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds.
method Original argument to estimate first and second variation of the area for isoperimetric sets, avoiding regularity theory.
result Generalizes results for smooth and non-compact manifolds, Alexandrov spaces, and convex bodies.
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…
The paper explores sharp isoperimetric properties on non-compact spaces with Ricci bounds.
problem Sharp isoperimetric properties on non-compact spaces with Ricci bounds.
method Sharp isoperimetric comparison theorems and asymptotic isoperimetric properties.
result Almost regularity theorems and enhanced functional inequalities.
In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. We then introduce a notion of "being asymptotically isoperimetric" for a family of finite a…
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.
We review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we …
We establish some a priori geometric relations on stable minimal surfaces lying inside three-manifolds with scalar curvature uniformly bounded below. The relations are based on a slight generalization of a formula due to Castillon. We apply it to prove non-local rigidity results in the particular sense that they expres…
Study on minimizing perimeter in unbounded convex bodies without boundary regularity.
problem Minimizing perimeter under volume constraint in unbounded convex bodies.
method Introducing uniform geometry, asymptotic cylinders, approximation, and approximation argument.
result Existence of isoperimetric regions in generalized sense and strict concavity of isoperimetric profile.
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.
The paper proves isoperimetric regions in specific manifolds cannot escape to infinity.
problem Exploring isoperimetric regions in asymptotically hyperbolic manifolds.
method Analyzing scalar curvature and Hawking mass to prove region exhaustivity.
result Isoperimetric regions with scalar curvature ≥ -6 cannot escape to infinity.
The paper characterizes general (α, β) norms in Minkowski spaces.
problem Characterizing general (α, β) norms in Minkowski spaces.
method By studying the behavior of Darboux curves and isoperimetric properties of indicatrix.
result Global geometric quantities that characterize Randers norms in arbitrary dimensions.
Study Finsler metric measure manifolds' concentration properties.
problem Understanding concentration properties in Finsler metric measure manifolds.
method Established relationships with observable diameter, isoperimetric inequalities, and first eigenvalue.
result Derived a Cheng type upper bound estimate for the first closed eigenvalue.
Researchers create manifolds with unusual isoperimetric properties.
problem Finding manifolds with discontinuous isoperimetric profiles.
method Constructing expander manifolds and 3-spheres.
result Discontinuous isoperimetric profiles exist in 2D Riemannian manifolds.
Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.
problem Proving geometric results for substatic Riemannian manifolds.
method Comparison theory based on a newly discovered conformal connection.
result Sharp, weighted Isoperimetric inequality quantifying boundary minimization.
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. Study minimal discs in metric spaces with a quadratic isoperimetric inequality.
problem Understanding the geometry of minimal discs in metric spaces.
method Associate a compact metric space to each minimal disc, controlling its properties by the isoperimetric inequality.
result The geometry of the associated space can control the shapes of curves and the original space's topology.
Score matching efficiency tied to distribution isoperimetric properties.
problem Understanding when score matching is as efficient as maximum likelihood.
method Connecting score matching efficiency to isoperimetric constants of distributions.
result Score matching is statistically efficient when the distribution has a small isoperimetric constant.
Study hyperbolic geometry to find Fibonacci numbers.
problem Investigate exponential growth and isoperimetric inequality in hyperbolic geometry.
method Combinatorial approximation of hyperbolic plane, calculations.
result Surprising link to Fibonacci numbers.
Minimal surfaces in hyperbolic space have a sharp area bound.
problem Bounding the renormalized area of minimal surfaces.
method Proving an inequality using conformal length of ideal boundary.
result Sharp isoperimetric property of renormalized area.
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.
A unique volume minimizer is found in a class of convex bodies.
problem Finding a unique minimizer for a reverse isoperimetric problem.
method Proving a reverse quermassintegral inequality.
result The convex hull of two balls is a unique minimizer among λ-concave bodies. We equip many non compact non simply connected surfaces with smooth Riemannian metrics whose isoperimetric profile is smooth, a highly non generic property. The computation of the profile is based on a calibration argument, a rearrangement argument, the Bol-Fiala curvature dependent inequality, together with new result…
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
Study optimizes perimeter in convex domains with anisotropic constraints.
problem Optimizing perimeter in convex domains with anisotropic constraints.
method Analytical properties, topological features, and geometric measure theory results.
result Sharp isoperimetric inequalities and existence of minimizers.
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.
In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve with a simple Fourier series proof: length, area of the region included by the …
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 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.
The study proves uniqueness of large isoperimetric sets in specific noncompact manifolds.
problem Proving uniqueness of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature.
method Analyzing properties of complete Riemannian manifolds with specific curvature conditions.
result There exists a set of volumes with density 1 at infinity where isoperimetric sets are unique and strictly volume preserving stable.
Study on isoperimetric inequalities and regularity of A-harmonic functions on surfaces.
problem Investigating isoperimetric inequalities and regularity of A-harmonic functions on smooth surfaces. method Logarithmic and power-type convexity of the length of level curves, higher Sobolev regularity properties, and estimates for derivatives.
result Higher Sobolev regularity properties of solutions, including W2,2 regularity. Random groups in the square model have Haagerup property and are residually finite.
problem Understanding properties of random groups in the square model.
method Generalized Isoperimetric Inequality and introduced modified hypergraphs to structure Cayley complex.
result Random groups in the square model have the Haagerup property and are residually finite.
Study spherical convex bodies using Lp-floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced Lp-floating areas and curvature entropy for spherical convex bodies. result Established isoperimetric inequalities and dual isoperimetric inequalities.
The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.
problem Investigating Gromov hyperbolicity of specific metrics.
method Using isoperimetric inequalities to characterize Gromov hyperbolicity.
result Characterization of domains where these metrics are Gromov hyperbolic.
Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every d there are unbounded degree simplicial co…
Study on electrostatic systems with boundary, proving new geometric inequalities.
problem Electrostatic systems with boundary in higher dimensions.
method Investigation of electrostatic systems on compact manifolds with boundary, establishing new geometric properties.
result Proved sharp boundary estimates and isoperimetric-type inequalities for electrostatic manifolds.
We prove that the isoperimetric profile of a convex domain Ω with compact closure in a Riemannian manifold (Mn+1,g) satisfies a second order differential inequality which only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of Ω. Regularity properties of the profile and top…
Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for Lφ affine surface areas are established.
Sharp density threshold for Property (T) found in random groups models.
problem Density threshold for Kazhdan's Property (T) in random groups.
method Quotient of free groups by random reduced words, new geometrical tools.
result Sharp density threshold for Property (T) equals 1/3.
For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanif…
Study shows convergence rates for Cheeger cuts on data clouds.
problem Optimizing graph cuts for clustering data sampled from a manifold.
method Analyzes statistical properties of Cheeger cuts on proximity graphs built from data.
result Obtains high probability convergence rates for Cheeger constant and cuts.