The p-widths of a polygon

2025/05/05 by Chodosh, Otis, Cholsaipant, Sithipont · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.03047

Abstract

The p-widths are a nonlinear analogue of the spectrum of the Laplacian. We prove that each p-width of a polygon in ℝ² is achieved by a union of billiard trajectories. We also compute the p-widths of the equilateral triangle for p=1,…,4 and square for p=1,…,3.

Citations

• Boundary behavior of limit-interfaces for the Allen-Cahn equation on Riemannian manifolds with Neumann boundary condition
• Strong multiplicity one theorems and homological min-max theory
• The first width of non-negatively curved surfaces with convex boundary
• A free boundary minimal surface via a 6-sweepout
• On continuous billiard and quasigeodesic flows characterizing alcoves and isosceles tetrahedra
• The first p-widths of the unit disk
• On the Multiplicity One Conjecture in Min-max theory
• Existence of infinitely many minimal hypersurfaces in closed manifolds
• Min-max theory for constant mean curvature hypersurfaces
• The Allen-Cahn equation on closed manifolds
• Weyl law for the volume spectrum
• The Width of Ellipsoids
• Morse index of multiplicity one min-max minimal hypersurfaces
• Min-max for phase transitions and the existence of embedded minimal\n hypersurfaces
• Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
• Minimax problems related to cup powers and Steenrod squares

Cited by

• Billiards and Hofer's Geometry
• Volume spectrum of fiber bundles and the widths of Berger spheres

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• Polygonal billiards and "optical tori"
• Billiards in a circle with trajectories circumscribing a triangle
• Outer length billiards on polygons
• Is every triangle a trajectory of an elliptical billiard?
• The maximum number of cycles in a triangular-grid billiards system with a given perimeter