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The p-widths of a polygon

Otis Chodosh, Sithipont Cholsaipant

TL;DR

This work establishes that for a convex polygon $P\subset\mathbb{R}^2$, each $p$-width $\\omega_p(P)$ is realized by a finite union of billiard trajectories with total length $\\sum_j\mathrm{length}(\\gamma_{p,j})$, connecting the Gromov–Guth min-max framework to polygonal billiards. The authors prove a polygonal billiard theorem using a reflection argument on the Allen–Cahn phase-transition regularization, yielding a decomposition of the limiting energy into straight line segments that trace out billiard paths. They further refine this picture in the equilateral triangle by showing billiards unfold to straight lines on a tiling of the plane, and compute explicit low-width values for both the equilateral triangle and the square (e.g., $\\omega_1(T)=\\omega_2(T)=\\tfrac{3}{2}$, $\\omega_3(T)=\\tfrac{3\sqrt{3}}{2}$, $\\omega_4(T)=3$, and $\\omega_1(S)=\\sqrt{2}$, $\\omega_2(S)=2$, $\\omega_3(S)=2\\sqrt{2}$). These results illuminate the structure of $p$-widths in polygonal domains and connect minimal-surface-type variational theories with classical billiard dynamics in planar geometry.

Abstract

The $p$-widths are a nonlinear analogue of the spectrum of the Laplacian. We prove that each $p$-width of a polygon in $\mathbb{R}^2$ is achieved by a union of billiard trajectories. We also compute the $p$-widths of the equilateral triangle for $p=1,\dots,4$ and square for $p=1,\dots,3$.

The p-widths of a polygon

TL;DR

This work establishes that for a convex polygon , each -width is realized by a finite union of billiard trajectories with total length , connecting the Gromov–Guth min-max framework to polygonal billiards. The authors prove a polygonal billiard theorem using a reflection argument on the Allen–Cahn phase-transition regularization, yielding a decomposition of the limiting energy into straight line segments that trace out billiard paths. They further refine this picture in the equilateral triangle by showing billiards unfold to straight lines on a tiling of the plane, and compute explicit low-width values for both the equilateral triangle and the square (e.g., , , , and , , ). These results illuminate the structure of -widths in polygonal domains and connect minimal-surface-type variational theories with classical billiard dynamics in planar geometry.

Abstract

The -widths are a nonlinear analogue of the spectrum of the Laplacian. We prove that each -width of a polygon in is achieved by a union of billiard trajectories. We also compute the -widths of the equilateral triangle for and square for .
Paper Structure (21 sections, 26 theorems, 48 equations, 4 figures)

This paper contains 21 sections, 26 theorems, 48 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The billiard conjecture
  3. The first width of a convex polygon
  4. Low widths of certain regular polygons
  5. Organization
  6. Acknowledgments
  7. Preliminaries
  8. Gromov--Guth $p$-widths
  9. Double-well phase transition theory
  10. Comparison between the min-max theories
  11. Proof of the billiard theorem
  12. Improving the billiard theorem for the equilateral triangle
  13. The first width of a convex polygon
  14. Low $p$-widths of the equilateral triangle
  15. $w_2(T)$
  16. ...and 6 more sections

Key Result

Theorem 1.3

For $P\subset \mathbb{R}^2$ a (compact) convex polygon and $p=1,2,\dots$ there's a finite set of billiard trajectories $\gamma_{p,1},\dots,\gamma_{p,N(p)} \subset P$ (possibly with repetition) so that

Figures (4)

  • Figure 1: The definition of the map $\phi : \partial T\times \partial T \to \mathcal{Z}_1(T,\partial T)$ used to construct an optimal $2$-sweepout of the equilateral triangle $T$.
  • Figure 2: Deriving an upper bound for the length of $\phi$ when $l_1\cap l_2 = p \in T$.
  • Figure 3: Deriving an upper bound for the length of $\phi$ when $l_1\cap l_2 = p$ lies outside of $T$ but on the same side of $T$ as $p_1$.
  • Figure 4: Deriving an upper bound for the length of $\phi$ when $l_1\cap l_2 = p$ lies outside of $T$ on the opposite side of $T$ as $p_1,p_2$.

Theorems & Definitions (55)

  • Conjecture 1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 45 more