The Isospectral Problem for p-widths: An Application of Zoll Metrics

We pose the isospectral problem for the $p$-widths: Is a riemannian manifold $(M^n,g)$ uniquely determined by its $p$-widths, $\{{\omega }_p(M,g){\}}_{p=1}^{\mathrm{\infty }}$? We construct many counterexamples on $S^2$ using Zoll metrics and the fact that geodesic $p$-widths are given by unions of immersed geodesics.