The p-widths of a polygon

TL;DR

This paper investigates the p-widths of polygons, showing they are achieved by billiard trajectories and explicitly computing these widths for specific polygons like equilateral triangles and squares.

Contribution

It establishes that polygon p-widths are realized by billiard trajectories and provides explicit calculations for certain polygons.

Findings

p-widths of polygons are achieved by billiard trajectories

Explicit p-widths computed for equilateral triangle and square

Demonstrates the nonlinear analogue of Laplacian spectrum

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 $p=1,\dots,3$.