The Gromov-Hausdorff Distance Between Consecutive Spheres

Abstract

We determine the Gromov-Hausdorff distance between consecutive unit round spheres equipped with their geodesic metrics. Put ${\zeta }_n:=\arccos (-\frac{1}{n+1}),$ the common geodesic distance between distinct vertices of a regular simplex with $n+2$ vertices inscribed in ${\mathbb{S}}^n$. We prove that $$d_{\mathrm{GH}}({\mathbb{S}}^n,{\mathbb{S}}^{n+1})=\frac{{\zeta }_n}{2}(n\ge 1),$$ resolving a conjecture of Lim, Mémoli, and Smith. All cases $n\ge 4$ were previously open. This equality is established by explicitly constructing a family of correspondences ${\mathcal{R}}_n\subseteq {\mathbb{S}}^{n+1}\times {\mathbb{S}}^n$, whose distortion matches the known quantitative Borsuk-Ulam lower bound ${\zeta }_n$. We also introduce synchronized spherical joins and suspensions of correspondences and prove that the distortion of a join is exactly the maximum of the distortions of its factors. In particular, suspension preserves distortion. Applying these join and suspension operations to the optimal correspondences ${\mathcal{R}}_n$ yields new bounds for spheres of nonconsecutive dimensions, including $$\underset{m\to \infty }{\lim }{d}_{\mathrm{GH}}({\mathbb{S}}^m,{\mathbb{S}}^{m+d(m)})=\frac{\pi }{4}\text{whenever }d(m)\ge 1,\text{and }d(m)=o(m).$$