arXiv research

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…

The study examines stable regions in weighted manifolds with boundary properties.

Given a positive lower semi-continuous density $f$ on $\mathbb{R}^2$ the weighted volume $V_f:=f\mathscr{L}^2$ is defined on the $\mathscr{L}^2$-measurable sets in $\mathbb{R}^2$. The $f$-weighted perimeter of a set of finite perimeter $E$ in $\mathbb{R}^2$ is written $P_f(E)$. We study minimisers for the weighted isop…

This paper solves a variation of the isoperimetric problem in higher dimensions.

Minimal partitions with minimal perimeter found in metric spaces.

New isoperimetric inequalities in the plane with radial weights identified.

Hexagonal tilings minimize perimeter with unequal volumes.

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

Study sharp inequalities for perimeter functionals in capillarity and convex cones.

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.

This study proves the Half Space Property for RCD(0,N) and RCD(K,N) spaces.

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…

Study shows zero level sets of solutions to Allen-Cahn equation are minimal surfaces with zero mean curvature.

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 establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an $(n-2)$-dimensional plane at $120^{\circ}$ …

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 some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to…

We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into $q$ cells of prescribed (positive) Gaussian measure when $2 \leq q \leq n+1$, is to use a "simplicial cluster", obtained from the Voronoi cells of $q$ equidistant points. Moreover, we prove that…

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 …

The paper proves inequalities for star-shaped and $F$-mean convex hypersurfaces in $\mathbb{R}^{n+1}$.

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.