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 S¹ and the n-dimensional sphere Sⁿ (for any n) when endowed with their respective geodesic metrics. Additionally, we prove that the GH distance between S³ and S⁴ is equal to 1/2(-1/4), thus settling the case $n=3$ of a conjecture by Lim, M\'emoli and Smith.