The p-widths of a polygon

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 $p=1,\dots ,3$.