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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 $0 \leq n \leq m$, $$d(\mathbb{S}_m,\mathbb{S}_n)\leq d(point,S_m)+d(point,S_n)\leq 2$$ But I find it difficult to control the lower bound with the inf over all possible metric spaces $M$.

I conjecture that for all $0 \leq n \leq m$: $$d(\mathbb{S}_m,\mathbb{S}_n)\geq \lambda_{m,n}\frac{m-n}{m}, \text{ where } \liminf_{m,n\to \infty}\lambda_{m,n}>0$$

I only know the Gromov-Hausdorff theory from Peterson's Riemannian Geometry, which does not give enough information to compute this distance. I will appreciate any pointers.

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.

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 $m=2$, $n=3$, 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 $0 \leq n \leq m$, $$d(\mathbb{S}_m,\mathbb{S}_n)\leq d(point,S_m)+d(point,S_n)\leq 2$$ But I find it difficult to control the lower bound with the inf over all possible metric spaces $M$.

I conjecture that for all $0 \leq n \leq m$: $$d(\mathbb{S}_m,\mathbb{S}_n)\geq \lambda_{m,n}\frac{m-n}{m}, \text{ where } \liminf_{m,n\to \infty}\lambda_{m,n}>0$$

I only know the Gromov-Hausdorff theory from Peterson's Riemannian Geometry, which does not give enough information to compute this distance. I will appreciate any pointers.

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 $0 \leq n \leq m$, $$d(\mathbb{S}_m,\mathbb{S}_n)\leq d(point,S_m)+d(point,S_n)\leq 2$$ But I find it difficult to control the lower bound with the inf over all possible metric spaces $M$.

I conjecture that for all $0 \leq n \leq m$: $$d(\mathbb{S}_m,\mathbb{S}_n)\geq \lambda_{m,n}\frac{m-n}{m}, \text{ where } \liminf_{m,n\to \infty}\lambda_{m,n}>0$$

I only know the Gromov-Hausdorff theory from Peterson's Riemannian Geometry, which does not give enough information to compute this distance. I will appreciate any pointers.

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: $d_{G-H}(\mathbb{S}_2,\mathbb{S}_3)\leq \sqrt{2}$. And in general case we have $d_{G-H}(\mathbb{S}_m,\mathbb{S}_n)\leq d_{G-H}(point,S_m)+d_{G-H}(point,S_n)\leq 2,\forall 0\leq n\leq m$. 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 $d_{G-H}(\mathbb{S}_m,\mathbb{S}_n)\geq \lambda_{m,n}\frac{m-n}{m},\forall 0\leq n\leq m$, 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.

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 $m=2$, $n=3$, 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 $0 \leq n \leq m$, $$d(\mathbb{S}_m,\mathbb{S}_n)\leq d(point,S_m)+d(point,S_n)\leq 2$$ But I find it difficult to control the lower bound with the inf over all possible metric spaces $M$. 

I conjecture that for all $0 \leq n \leq m$: $$d(\mathbb{S}_m,\mathbb{S}_n)\geq \lambda_{m,n}\frac{m-n}{m}, \text{ where } \liminf_{m,n\to \infty}\lambda_{m,n}>0$$

I only know the Gromov-Hausdorff theory from Peterson's Riemannian Geometry, which does not give enough information to compute this distance. I will appreciate any pointers.