arXiv research

Minimal partitions with minimal perimeter found in metric spaces.

Hexagonal tilings minimize perimeter with unequal volumes.

Perimeter on manifolds leads to new symmetrization methods.

We show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional.

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

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

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…

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

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

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

Study optimizes perimeter in convex domains with anisotropic constraints.

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 …

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

New geometric insights reveal properties of adversarial training problems.

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

Study proves rigidity of critical points in hydrophobic capillary systems.

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 sharp inequalities for perimeter functionals in capillarity and convex cones.

We describe the configuration space $\mathbf{S}$ of polygons with prescribed edge slopes, and study the perimeter $\mathcal{P}$ as a Morse function on $\mathbf{S}$. We characterize critical points of $\mathcal{P}$ (these are \textit{tangential} polygons) and compute their Morse indices. This setup is motivated by a num…

We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms $G_p: T_pM \to [0,\infty]$ are given. When we consider sub-Riemannian manifolds, our definition coincide with the one given in the more general context of metric measure spaces which are doub…

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.

Study non-local isoperimetric energies on spheres using a Riemannian autocorrelation function.

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 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…

Optimally regularizes boundaries in the Heisenberg group with prescribed curvature.

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…

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

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.

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…

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

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

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…

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…

Study on shapes in Heisenberg group with convex body norms.

The study proves properties of optimizers for sets maximizing perimeter under fixed volume constraints.

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.

Here a new notion of fractional length of a smooth curve, which depends on a parameter $σ$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length u…

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

New mathematical surfaces without boundaries found.