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

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, Mémoli and Smith.