# How to compute the Gromov-Hausdorff distance between spheres $latex S_n$ and $latex S_m$?

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 $latex \mathbb{S}_n \to \mathbb{R}^{n+1}$. Is it possible for us to compute the Gromov-Hausdorff distance $latex d_{G-H}(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $latex \mathbb{S}_n$ and $latex \mathbb{S}_m$, $latex m\neq n$?

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

At least we can embed $latex \mathbb{S}_2$,$latex \mathbb{S}_3$ into $latex \mathbb{R}^3$ in a canonical way. This will lead to a upper bound: $latex d_{G-H}(\mathbb{S}_2,\mathbb{S}_3)\leq \sqrt{2}$. And in general case we have $latex d_{G-H}(\mathbb{S}_m,\mathbb{S}_n)\leq d_{G-H}(point,S_m)+d_{G-H}(point,S_n)\leq 2,\forall 0\leq n\leq m$. But it is difficult to get a lower bound control for me. Because we need to take the inf in all possible metric spaces $latex M$. Especially I conjecture $latex d_{G-H}(\mathbb{S}_m,\mathbb{S}_n)\geq \lambda_{m,n}\frac{m-n}{m},\forall 0\leq n\leq m$, where $latex \liminf_{m,n\to \infty}\lambda_{m,n}>0$.

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.

And we know for the case $latex S_n,S_m$, if $latex n,m$ is very near to each other,then the two space should be more near, and there is a canonical embed $latex S_0\subset S_1 \subset S_2 ….\subset S_n \subset …$. So it is natural to conjecture if $latex m,n$ is very near then the distance $latex d_{G-H}(S_n,S_m)$ is very small. I have a very rough strategy to prove the conjecture, that is inspired by the Nash embedding theorem. I just mean if we consider the problem in this frame $latex d_{G-H}(S_n,S_m)=\inf_{M,g,f}(d_M(f(S_n),g(S_m)))$ then the difficult is the deformation space of $latex M,g,f$ is too large. so the first step is to establish a regular lemma, to prove the function $latex d_M(f(S_n),g(S_m))$ is continues under the small perbutation of $latex M$ and reduced to the situation of space $M,g,f$ with very nice regularity. the second part is to embed $latex M$ to a big euclid space $latex R^N$ as subspace, and the embedding stay the length of geodesic.locally this is determine by a group of pde:$latex u_i(x)u_j(x)=g_{ij}(x)$,at least in the cut locus.but there should be some critical point,and I do not know how to deal with them.the third,i.e. the last step is to calculate $latex d_{G-H}(S_n,S_m)$ in the very some deformation space $latex M,f,g$.

@Mark Sapir,Appreciate for help!I am reading the article you point out,it seems this article mainly focus on investigating the Gromov-Hausdorff limit space of a sequence of hyperbolic group equipped with modified G-H metric defined in 2.A with some special condition to ensure the limit space exists.and take a sequences corvarage to the limit space,the hyperbolic property and some other thing is stayed by the process of take limit.
@Mark Sapir,So it is natural for us to investigate the original space by some information from the limit space.there is a series of bi-product state in 3.B.but I do not see where the author exactly calculate some groom-hausdorff distance of two different space,may you point out it?appreciate again!
@MarianoSuárez-Álvarez,Corrected, thanks.

Y:
I fixed numerous typos. In particular, you should use spacing after each punctuation mark; capitals to begin sentences and names.

H:
Thank you very much for helping me to correct the mistakes! I will know how to write in a correct style.

Y:
23.1k
Your conjecture would imply that the GH distance is unbounded. But it’s clearly bounded, since the GH distance of any sphere to a point is equal to 2 (when the sphere is endowed with the restriction of Euclidean distance, as you seem to assume, or $latex \pi$ when endowed with geodesic distance) and hence the GH distance between any two spheres is $latex \le 4$.

H:
73
You are right,In fact if we use the canonical embed, then we can get $latex d_{G-H}(S_n,S_m)\leq 2$ by another equivalent definition of GH distance.I confuse the geometry picture of the pairs $latex T_n,S_n$ with the pairs $latex S_n,S_m$,for $latex S_n,S_m$ case,I thick the seems correct conjecture will be $latex d_{G-H}(S_n,S_m)\sim \frac{m-n}{m},0\leq n\leq m,m,n\to \infty$.

Y:
16:37
Clearly from standard embeddings we get $latex d_{GH}(S_n,S_m)\le\sqrt{2}$ for all $latex n,m\ge 0$. Would it be reasonable to simply conjecture that it’s an equality whenever $latex n\neq m$?

H:
73
Yeah, you are right,$latex d_{G-H}(S_n,S_m)\leq \sqrt{2}$ for all $latex n,m\geq 0$.I find the interesting problem when I want to find a toy model of a kind of problem,roughly speaking is to investigate a map $latex f:X\to Y$ from low-dimensions space $latex X$ to high-dimension space $latex Y$ stay some affine structure of the low-dimension space $latex X$. This structure could have some control by the distance function on the low-dimension space, so if we can get some control on the variation of the Energy of distance function, this will share some line on the original problem I consider.
And we know for the case $latex S_n,S_m$, if $latex n,m$ is very near to each other,then the two space should be more near, and there is a canonical embed $latex S_0\subset S_1 \subset S_2 ….\subset S_n \subset …$. So it is natural to conjecture if $latex m,n$ is very near then the distance $latex d_{G-H}(S_n,S_m)$ is very small. .
I have a very rough strategy to prove the conjecture, that is inspired by the Nash embedding theorem. I just mean if we consider the problem in this frame $latex d_{G-H}(S_n,S_m)=\inf_{M,g,f}(d_M(f(S_n),g(S_m)))$ then the difficult is the deformation space of $latex M,g,f$ is too large. so the first step is to establish a regular lemma, to prove the function $latex d_M(f(S_n),g(S_m))$ is continues under the small perbutation of $latex M$ and reduced to the situation of space $latex M,g,f$ with very nice regularity.
The second part is to embed $latex M$ to a big euclid space $latex R^N$ as subspace, and the embedding stay the length of geodesic.locally this is determine by a group of pde:$latex u_i(x)u_j(x)=g_{ij}(x)$,at least in the cut locus.but there should be some critical point,and I do not know how to deal with them.the third,i.e. the last step is to calculate $latex d_{G-H}(S_n,S_m)$ in the very some deformation space $latex M,f,g$.
I need come back to explain why we expect the groom-hausdorff distance $latex d_{G-H}(S_d,S_m),0\leq n\leq m$ should be much small than $latex \sqrt 2$ when $latex frac{n}{m}$ is small.
Let consider a toy model of the problem,in a graph model,i.e. now we do not consider to take the Infimum in all space but in discrete space endow with metric. this can be view as a complete graph equipped metric, i.e. $latex M=\{(G,d_G)\}$. So there is also some space very like $latex S_n$,$S_m$ in the Euclid space, Let remark them as $latex G_{S_n},G_{S_m}$.
, oberseve that $latex (G,d_G)\in M$ then $latex (G,\hat d_{G})\in M$,$latex \hat d_{G}$ is a scaling of $latex d_G$so it is natural to consider a cut off of $latex M$,called $latex M_{\lambda}$ which is just a subset of $latex M$ and satisfied if $latex (G,d_G)\in M_{\lambda}$,then $latex \inf_{x\neq y}d_{G}(x,y)\geq d$.
Now,in the space $latex M_{\lambda}$ let us consider a Distance distribution:$latex \mu_G((a,b))=\frac{\#\{x,y\in G|a<d_G(x,y)<b\}}{\#G\times G}$. Then this distribution will give us some information of the distance of the two different set $latex G_1,G_2$ in $latex M_{\lambda}$.
(removed)
Now,in the space $latex M_{\lambda}$ let us consider a Distance distribution:$latex \mu_G((a,b))=\frac{\#\{x,y\in G|a<d_G(x,y)<b\}}{\#G\times G}$. Then this distribution will give us some information of the distance of the two different set $latex G_1,G_2$ in $latex M_{\lambda}$.
and obviously we will see that if $latex n,m$ is close,then the distribution of fuzzy approximation $latex G_{S_n},G_{S_m}$ is near, and the reverse is also true. I think this can explain why the conjecture $latex d_{G-H}(S_n,S_m) =O(\frac{m-n}{m})$ may be right.

by the way,it is a very good exercise to proof $latex d_{G-H}(S_n,S_0)=1,\forall n\in N^*$.