We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,181 papers · 148 categories
Trend · papers per month
10 results for “least-perimeter”
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.
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.
Planar Soap Bubblesmath.MG
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…
In 1D, optimal double bubbles are intervals or spheres.
problem Finding the least-perimeter way to enclose two volumes with a log-convex density.
method Analyzing the density function's log-convexity to determine the optimal configuration.
result In 1D, the optimal configuration can be intervals or spheres.
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.
The study confirms two cases of the convex body isoperimetric conjecture in the plane.
problem The least perimeter to enclose a given area inside a unit disk is greater than inside any other convex set.
method Examined symmetric domains and perturbations of the unit disk.
result Two cases of the convex body isoperimetric conjecture are confirmed.
The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.
problem Finding the minimum perimeter for polygons with a fixed area in hyperbolic geometry.
method Proving analogues of the discrete isoperimetric inequality for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons.
result Established two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with certain area or perimeter restrictions.
Researchers prove the least perimeter way to divide space into three parts.
problem Finding the least perimeter way to divide space into three parts.
method Proved using Gaussian measure and isoperimetric minimizers.
result Tripod-clusters are the unique minimizers for dividing space into three parts.
Proves least Gaussian perimeter decomposition conjectures for 2-3 cells in n-dimensional space.
problem Finding least perimeter ways to divide space into cells of prescribed Gaussian measure.
method Analyzes stable clusters and uses Voronoi cells of equidistant points.
result Simplicial clusters are unique minimizers for 2-3 cells in n-dimensional space.