Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.
problem Understanding the structure of perimeter minimizing sets in specific metric spaces.
method Established a monotonicity formula and proved rigidity for perimeter minimizers in RCD(0,N) spaces.
result Sharp Hausdorff dimension estimates for singular strata and existence of blow-down cones.
Perimeter minimizers in curved spaces have a singular set no more than 5 dimensions.
problem Understanding the structure of minimizers in spaces with bounded Ricci curvature.
method Analysis of non-collapsed Ricci limit spaces with two-sided curvature bounds.
result The Hausdorff dimension of the singular set is at most \(N-5\).
Study fractional Allen-Cahn equation and nonlocal minimal surfaces, improving energy and perimeter estimates.
problem Properties of solutions to fractional Allen-Cahn equation and stationary nonlocal minimal surfaces.
method Quantitative stratification principle applied to fractional Allen-Cahn equation, leading to optimal estimates.
result Sharp potential energy and perimeter estimates for fractional Allen-Cahn equation and nonlocal minimal surfaces.
Investigates convexity of minimizers under mass constraint using nonlocal perimeter and potential.
problem Convexity of minimizers under mass constraint.
method Nonlocal free energy with nonlocal perimeter and convex potential.
result Quantitative stability theorem for nonlocal free energy assuming symmetry on the potential.
Minimal partitions with minimal perimeter found in metric spaces.
problem Finding minimal partitions with minimal perimeter in metric spaces.
method Existence proof and regularity analysis of minimal domains.
result Existence and regularity of minimal partitions in various metric spaces.
DUPLE tackles cross-deployment recognition in fiber-optic perimeter security with meta-learning.
problem Cross-deployment recognition challenges in fiber-optic perimeter security due to label scarcity and distribution shifts.
method DUPLE employs statistically guided meta-learning to enhance recognition robustness across unseen deployments.
result DUPLE consistently outperforms traditional and meta-learning baselines in cross-deployment DFOS benchmarks.
Hexagonal tilings minimize perimeter with unequal volumes.
problem Finding optimal tessellations with unequal cell volumes.
method Minimizing perimeter functionals for different classes of problems.
result Hexagonal tilings are optimal among partitions with almost equal areas.
This note develops certain sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter.
Infinite hyperbolic manifolds share same perimeter-to-volume ratio.
problem Finding hyperbolic manifolds with a fixed perimeter-to-volume ratio.
method Constructing infinitely many hyperbolic manifolds with nonempty boundaries.
result Existence of incommensurable hyperbolic manifolds with a fixed perimeter-to-volume ratio.
The paper examines perimeter minimizing sets in curved spaces and finds conditions for their boundary to match a specific structure.
problem Conditions for perimeter minimizing sets in curved spaces to have a boundary matching a product structure.
method Analyzes Riemannian manifolds with non-negative sectional curvature and quadratic volume growth.
result The boundary of a perimeter minimizing set in such manifolds is identified with a slice in the product structure.
Study sharp inequalities for perimeter functionals in capillarity and convex cones.
problem Quantitative isoperimetric inequalities for perimeter functionals in capillarity and convex cones.
method Derivation of Fuglede-type estimates and application of selection principle.
result Sharp quantitative isoperimetric inequalities in strong and barycentric forms.
Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.
problem Finding sets of least perimeter in Riemannian manifolds.
method Short proof using de Giorgi and Miranda's tradition in flat space.
result Reduced boundary of least perimeter sets is a smooth minimal hypersurface in low dimensions.
Study minimizers of quasi-perimeters in RCD spaces with volume constraints.
problem Regularity and topological properties of volume constrained minimizers in RCD spaces.
method New Deformation Lemma and study of interior and exterior points.
result Volume constrained minimizers are open bounded sets with Ahlfors regular boundary.
Perimeter on manifolds leads to new symmetrization methods.
problem Applying symmetrization methods to quasilinear elliptic problems on RN. method Generalization of perimeter to manifolds, using hear kernel regularization.
result New symmetrization method on spheres for quasilinear elliptic problems.
The paper studies properties of RCD(K,N) spaces and their boundaries.
problem Understanding the boundary structure and unit normal on RCD(K,N) spaces. method Proves concentration of boundary measure, discusses localization of unit normal, and develops tools for perimeter minimizers.
result Proves that the boundary measure of sets with finite perimeter is concentrated on the n-regular set Rn. Flow preserves volume on flat torus, converging to stable set.
problem Volume preservation in discrete mean curvature flow on flat torus.
method Discrete mean curvature flow, quantitative Alexandrov estimate, characterization in 2D.
result Flow converges exponentially fast to stable set.
The generalized soap bubble problem seeks the least perimeter way to enclose and separate n given volumes in R^m. We study the possible configurations for perimeter minimizing bubble complexes enclosing more than two regions. We prove that perimeter minimizing planar bubble complexes with equal pressure regions and wit…
We show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional.
We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapion…
Investigates dual foliations of polygon spaces based on area and perimeter.
problem Understanding dual foliations of polygon spaces guided by area and perimeter.
method Investigated topology of leaves, determined homology groups, and extended isoperimetric duality.
result Homology groups and homotopy types of polygon spaces are determined.
We describe all families of star-shaped n-polygons in the Euclidean plane with prescribed perimeter and area ; they are leaves of a foliation F on the space of star-shaped n-polygons. By the way, we study some geometric properties of convex polygons, for instance their inscriptibility in a circle and their regularity i…
We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.
We prove a monotonicity result at specific points for the Horizontal Perimeter for a class of surfaces in the Heisenberg group.
Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.
problem Regularity of boundaries in plentiful groups.
method Lipschitz approximation of boundaries.
result Boundary of almost minimizers can be approximated by intrinsic Lipschitz graphs.
We prove the existence of a perimeter-minimizing partition of R^n into regions of unit volume. We conclude with a short tribute to the late Manuel A. Fortes.
We prove that a family of entire intrinsic minimal graphs in the Heisenberg group are not perimeter minimizing.
A lens cluster minimizes perimeter in the plane with given area constraints.
problem Minimizing perimeter in the plane with given area constraints.
method Analyzing lens clusters consisting of circular arcs with specific geometric properties.
result Lens clusters are local minimizers of the total perimeter functional.
The paper explores connections between perimeter, area, and visual angle of convex sets.
problem Understanding geometric properties of convex sets through visual angle and related measurements.
method Establishing universal formulas and characterizing convex sets of constant width.
result Crofton's formula is the unique universal formula relating visual angle, length, and area.
We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of J.-M.\ Schlenker.
Study fractional perimeter asymptotics on Riemannian manifolds as s approaches 0.
problem Asymptotics of fractional perimeter on Riemannian manifolds.
method Analysis of fractional Laplacian and existence of bounded harmonic functions.
result Asymptotics of fractional s-perimeter on all complete manifolds. We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H^2 we get a formula that appears to be new. We show that it implies the standard Cauchy-Santalo formula involving a central ang…
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 proves existence of regions minimizing perimeter in specific geometric structures.
problem Existence of isoperimetric regions in sub-Finsler nilpotent groups.
method Analyzes nilpotent Lie groups with a bracket-generating distribution and asymmetric norms.
result Proves existence of minimizers of perimeter under volume constraint.
Maximising the detection of intrusions is a fundamental and often critical aim of perimeter surveillance. Commonly, this requires a decision-maker to optimally allocate multiple searchers to segments of the perimeter. We consider a scenario where the decision-maker may sequentially update the searchers' allocation, lea…
This study proves the Half Space Property for RCD(0,N) and RCD(K,N) spaces.
problem Proving the Half Space Property for RCD(K,N) spaces.
method Analyzing locally perimeter minimizing sets and extending Green's functions results.
result The Half Space Property holds for RCD(K,N) spaces under specific conditions.
The hypercube's perimeter is significantly larger than expected near half volume.
problem Understanding the isoperimetric profile of the hypercube.
method Analytical proof of perimeter bounds and comparison to Gaussian isoperimetric profile.
result The isoperimetric profile of the hypercube does not converge to the Gaussian profile as dimension increases.
Study finds critical points in perimeter functional for fixed volume sets.
problem Finding critical points in perimeter functional for sets of fixed volume.
method Utilizes Mazurwoski--Zhou techniques and new Cacciopoli set connectedness results.
result Constructs smooth almost embedded hypersurfaces with non-zero constant mean curvature.
New mathematical surfaces without boundaries found.
problem Existence of nonlocal free boundary minimal surfaces.
method Fractional perimeter critical points with invariant boundary.
result Existence of nonlocal free boundary minimal surfaces without boundaries.
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.
This note is devoted to the study of sets of finite perimeter over RCD(K,N) metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of …
We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in Rn. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity…
Given an elliptic integrand of class C3, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.
We prove that the unique least-perimeter way of partitioning the unit 2-dimensional disk into three regions of prescribed areas is by means of the standard graph consisting in three balanced constant geodesic curvature curves meeting themselves at 120 degrees, and reaching orthogonally the boundary of the disk.
Develops BV function and finite perimeter set theory on Riemannian manifolds.
problem Theory of BV functions and finite perimeter sets on arbitrary Riemannian manifolds.
method Localization framework combining Euclidean and metric measure space techniques.
result Recovery of key Euclidean results in Riemannian setting.
This work is an investigation of perimeter measures in the metric measure space given by the Heisenberg group with Haar measure and a Carnot-Carathéodory metric, which is in general a sub-Finsler metric. Included is a reduction of Minkowski content in any CC-metric to an integral formula in terms of Lebesgue surface ar…
New proofs and inequalities for capillarity problems quantify asymmetries.
problem Quantifying asymmetries in capillarity functionals.
method ABP-type technique, symmetrization, selection-type argument.
result Sharp quantitative inequalities for asymmetries in capillarity problems.
We consider the configuration space of planar n-gons with fixed perimeter, which is diffeomorphic to the complex projective space CPn−2. The oriented area function has the minimal number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute …
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 …