arXiv research

Minimal partitions with minimal perimeter found in metric spaces.

Hexagonal tilings minimize perimeter with unequal volumes.

Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.

Infinite hyperbolic manifolds share same perimeter-to-volume ratio.

The paper examines perimeter minimizing sets in curved spaces and finds conditions for their boundary to match a specific structure.

Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.

Study minimizers of quasi-perimeters in RCD spaces with volume constraints.

Perimeter on manifolds leads to new symmetrization methods.

The paper studies properties of $\mathrm{RCD}(K,N)$ spaces and their boundaries.

Perimeter minimizers in curved spaces have a singular set no more than 5 dimensions.

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…

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.

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.

DUPLE tackles cross-deployment recognition in fiber-optic perimeter security with meta-learning.

A lens cluster minimizes perimeter in the plane with given area constraints.

The paper explores connections between perimeter, area, and visual angle of convex sets.

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.

Two natural foliations, guided by area and perimeter, of the configurations spaces of planar polygons are considered and the topology of their leaves is investigated in some detail. In particular, the homology groups and the homotopy type of leaves are determined. The homology groups of the spaces of polygons with fixe…

Study fractional perimeter asymptotics on Riemannian manifolds as $s$ approaches 0.

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.

Study proves existence of regions minimizing perimeter in specific geometric structures.

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…

The hypercube's perimeter is significantly larger than expected near half volume.

Study finds critical points in perimeter functional for fixed volume sets.

Investigates convexity of minimizers under mass constraint using nonlocal perimeter and potential.

New mathematical surfaces without boundaries found.

Study optimizes perimeter in convex domains with anisotropic constraints.

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 $\mathbb{R}^n$. 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 $\mathscr{C}^{3}$, 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.

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…

We consider the configuration space of planar $n$-gons with fixed perimeter, which is diffeomorphic to the complex projective space $\mathbb{C}P^{n-2}$. 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 …

Study proves rigidity of critical points in hydrophobic capillary systems.

We present a new blow-up method that allows for establishing the first general formula to compute the perimeter measure with respect to the spherical Hausdorff measure in noncommutative nilpotent groups. This result leads us to an unexpected relationship between the area formula with respect to a distance and the profi…

Paper proves isoperimetric inequality for Minkowski spacetime.

New geometric insights reveal properties of adversarial training problems.

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some …

Study fractional Allen-Cahn equation and nonlocal minimal surfaces, improving energy and perimeter estimates.

We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to de…