There is the question, because when we consider the Gromov-Hausdorff distance, we must fix the metric, so we use the natural metric induced from the embedding $\mathbb{S}_n \to \mathbb{R}^{n+1}$. Is it possible for us to compute the Gromov-Hausdorff distance $d_{G-H}(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$, $m\neq n$?

For example if we want to calculate $d_{G-H}(\mathbb{S}_2,\mathbb{S}_3)=\inf_{M,f,g}d_{M}(\mathbb{S}_2,\mathbb{S}_3)$, where $M$ ranges over all possible metric space and $f:\mathbb{S}_2\to M$ and $g:\mathbb{S}_3\to M$ range over all possible isometric (distance-preserving) embeddings.

At least we can embed $\mathbb{S}_2$,$\mathbb{S}_3$ into $\mathbb{R}^3$ in a canonical way. This will lead to a upper bound: $d_{G-H}(\mathbb{S}_2,\mathbb{S}_3)\leq \sqrt{3}$. And in general case we have $d_{G-H}(\mathbb{S}_m,\mathbb{S}_n)\leq \sqrt{\max(m,n)}$. But it is difficult to get a lower bound control for me. Because we need to take the inf in all possible metric spaces $M$. Especially I conjecture $d_{G-H}(\mathbb{S}_m,\mathbb{S}_n)\geq \lambda_{m,n}\sqrt{\max(m,n)}$, where $\liminf_{m,n\to \infty}\lambda_{m,n}>0$.

I only know the knowledge of Gromov-Hausdorff from Peterson's Riemann Geometry. Unfortunately there is not enough information to compute the Gromov-Hausdorff distance, so this problem may be very stupid, I will appreciate any pointer.