Some novel constructions of optimal Gromov-Hausdorff-optimal correspondences between spheres

TL;DR

This paper provides new proofs for the Gromov-Hausdorff distances between spheres and confirms a specific conjecture for the case n=3, advancing understanding of metric geometry between spherical spaces.

Contribution

It offers alternative proofs for known GH distances between spheres and proves the exact GH distance between S^3 and S^4, settling a conjecture.

Findings

GH distance between S^1 and S^n determined

GH distance between S^3 and S^4 computed as 0.5 * arccos(-1/4)

Confirmed case n=3 of a conjecture by Lim, Mémoli, and Smith

Abstract

In this article, as a first contribution, we provide alternative proofs of recent results by Harrison and Jeffs which determine the precise value of the Gromov-Hausdorff (GH) distance between the circle $\mathbb{S}^1$ and the $n$-dimensional sphere $\mathbb{S}^n$ (for any $n\in\mathbb{N}$) when endowed with their respective geodesic metrics. Additionally, we prove that the GH distance between $\mathbb{S}^3$ and $\mathbb{S}^4$ is equal to $\frac{1}{2}\arccos\left(\frac{-1}{4}\right)$, thus settling the case $n=3$ of a conjecture by Lim, M\'emoli and Smith.