arXiv research

Optimal enclosures of four equal-area regions are connected.

We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.

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…

Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M,g)=1 and every 1-cycle dividing M into two regions of equal area has length >C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Ste…

The paper introduces new measures for ovals and finds geometric relations between their properties.

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 …

Consider a 2-plane $P \subset \mathbb{C}^n$ and let $D$ be a bounded region in $P$ with a piecewise-smooth boundary. Let $I(D)$ be the infimum of areas of all piecewise-smooth isotropic surfaces in $\mathbb{C}^n$ with the same boundary as $D$. Then $I(D)= λ_P^n \cdot Area(D)$. If $P$ is not complex, $λ_P^n < \frac{3π}{…

New set class preserves Fourier series terms for planar ovals, leading to isoperimetric inequalities.

Paper improves isoperimetric inequality with new curve properties.

The paper extends the isoperimetric inequality to disconnected regions.

Proposes a new algorithm for solving optimization problems with stochastic objectives and equality constraints.

We show some characterizations of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with intrinsic and extrinsic properties such as the $n$-dimensional area of the sections cut off by hyperplanes, the $(n+1)$-dimensional volume of regions between parallel hyperplanes, and the $n$-dimensional surf…

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygo…

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.

We prove that if $M$ is a three-manifold with scalar curvature greater than or equal to -2 and $Σ\subset M$ is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of $Σ$ is greater than or equal to $4π(g(Σ)-1)$, where $g(Σ)$ denotes the genus of $Σ$. In t…

In this work we investigate the following isoperimetric problem in the hyperbolic plane: to find the regions of prescribed area with minimal perimeter between two parallel horocycles. We give an explicit and detailed description of all such regions.

Researchers find minimal-area metrics on surfaces with constraints.

The Prytz planimeter is a simple example of a system governed by a non-holonomic constraint. It is unique among planimeters in that it measures something more subtle than area, combining the area, centroid and other moments of the region being measured, with weights depending on the length of the planimeter. As a tool …

Study estimates area of non-compact surfaces in 3-manifolds, proving rigidity under certain conditions.

Study improves precipitation predictions for High Mountain Asia using machine learning.

New Boolean algebra method shows knot unknotting number is (c+1)/2.

This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers…

P. Papasoglu asked in [Pap13] whether for any Riemannian 3-disk $M$ with diameter $d$, boundary area $A$ and volume $V$, there exists a homotopy $S_t$ contracting the boundary to a point so that the area of $S_t$ is bounded by $f(d,A,V)$ for some function $f$. He further asks whether it is possible to subdivide $M$ by …

Study proves Chern-Osserman type equality for surfaces in Euclidean space.

Simple proof for Cauchy's surface area formula.

Graph-based weather prediction adapted for local models.

Geotagged data can be used to describe regions in the world and discover local themes. However, not all data produced within a region is necessarily specifically descriptive of that area. To surface the content that is characteristic for a region, we present the geographical hierarchy model (GHM), a probabilistic model…

Study isoperimetric equalities for special closed curves.

New method aligns brain regions across subjects better than existing approaches.

The paper explores rigid geometric structures near surfaces with equality in area-charge inequalities.

Narrow neural networks have unbounded decision regions.

In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in the horospheres is not included). We reduce the problem to the study of rotationa…

Estimates the upper bound of linear regions in spheres centered at specific data points in ReLU neural networks.

Enhances DES by removing noise and defining regions more accurately.

The paper proposes a method to create density-equalizing maps for open surfaces.

In this work we classify the stable regions (second order minima of perimeter under an area constraint) in tori of revolution with piecewise continuous decreasing Gauss curvature from the longest parallel and with a horizontal symmetry. Some applications to isoperimetric problems are also given.

The paper finds singular isoperimetric regions in high-dimensional spaces.

This paper constructs Brownian motion on complex flag manifolds and finds joint distribution of stochastic areas.

Study on unknotting twisted knots using arc shift and region arc shift moves.

The paper establishes curvature inequalities and rigidity results for surfaces in Riemannian and Lorentzian geometry.

The paper establishes curvature inequalities and rigidity results for CMC and STCMC surfaces in Riemannian and Lorentzian geometry.

Graph embeddings from commute networks identify socioeconomic disparities in urban areas.

We show linear XOR classification is possible and propose equality separation for anomaly detection.

The paper confirms Yau's conjecture for minimal rotational hypersurfaces.

Study maps Spanish dialects on Twitter, revealing urban vs regional variations.

Area-charge inequalities and local rigidity of free boundary MOTS in charged initial data sets

Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.

Fragmented exchanges arise due to speed advantages in high-activity regions.