How to compute the Gromov-Hausdorff distance between spheres $S_n$ and $S_m$? Can we compute the Gromov-Hausdorff distance $d(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$, $m\neq n$? We consider the spheres with the metrics induced by the embedding
$\mathbb{S}_n \to \mathbb{R}^{n+1}$.
For $n=2$, $m=3$, this is evaluating $$d(\mathbb{S}_2,\mathbb{S}_3)=\inf_{M,f,g}d_{M}(\mathbb{S}_2,\mathbb{S}_3)$$ where $M$ ranges over all possible metric spaces and $f:\mathbb{S}_2\to M$ and $g:\mathbb{S}_3\to M$ range over all possible distance-preserving embeddings. As an upper bound, $d(\mathbb{S}_2,\mathbb{S}_3)\leq \sqrt{2}$.
More generally for $0 \leq n \leq m$,
$$d(\mathbb{S}_n,\mathbb{S}_m)\leq d(point,S_n)+d(point,S_m)\leq 2$$
But I find it difficult to control the lower bound with the inf over all possible metric spaces $M$.
I conjecture that for all $0 \leq n \leq m$:
$$d(\mathbb{S}_n,\mathbb{S}_m)\geq \lambda_{m,n}\frac{m-n}{m}, \text{ where } \liminf_{m,n\to \infty}\lambda_{m,n}>0$$
I only know the Gromov-Hausdorff theory from Petersen's Riemannian Geometry, which does not give enough information to compute this distance. I will appreciate any pointers.
 A: Hu Xiyu. Even though yours is a question from four years ago, I want to bring to your attention my recent paper “Gromov-Hausdorff distance between spheres”( https://arxiv.org/abs/2105.00611 ) coauthored with Facundo Memoli and Zane Smith, since it is very closely related to your question.
In this paper, we compute/bound the Gromov-Hausdorff distance between two spheres with different dimension (each with geodesic metric). We use topological methods in order to obtain lower bounds: more precisely, we resort to a certain version of Borsuk-Ulam Theorem for discontinous functions. On the other hand, we design specialized optimal correspondences in order to estimate upper bounds. In particular, we were able to compute precise value of the Gromov Hausdorff distance for $\mathbb{S}^1$ vs $\mathbb{S}^2$, $\mathbb{S}^1$ vs $\mathbb{S}^3$, and $\mathbb{S}^2$ vs $\mathbb{S}^3$.
Finally, the last section of the paper deals with the case of spheres with Euclidean metric. Even though we could not give a full answer, I believe you will be able to find some useful observations there.
