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 $\mathbb{S}n o \mathbb{R}^{n+1}$. 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 $\mathbb{S}_m$, $m
eq n$?
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 spaces and $f:\mathbb{S}_2 o M$ and $g:\mathbb{S}_3 o M$ 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 an upper bound: $d{G-H}(\mathbb{S}2,\mathbb{S}3) \leq \sqrt{2}$. And in general case we have $d{G-H}(\mathbb{S}m,\mathbb{S}n) \leq d{G-H}(\text{point},\mathbb{S}m) + d_{G-H}(\text{point},\mathbb{S}n) \leq 2, orall 0 \leq n \leq m$. 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 $d_{G-H}(\mathbb{S}m,\mathbb{S}n) \geq \lambda{m,n} rac{m-n}{m}, orall 0 \leq n \leq m$, where $\liminf{m,n o \infty}\lambda_{m,n} > 0$.
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 $S_n,S_m$, if $n,m$ is very near to each other, then the two space should be more near, and there is a canonical embed $S_0 \subset S_1 \subset S_2 \subset \ldots \subset S_n \subset \ldots$. So it is natural to conjecture if $m,n$ is very near then the distance $d_{G-H}(S_n,S_m)$ 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 $d_{G-H}(S_n,S_m)=\inf_{M,g,f}(d_M(f(S_n),g(S_m)))$ then the difficulty is the deformation space of $M,g,f$ is too large. So the first step is to establish a regular lemma, to prove the function $d_M(f(S_n),g(S_m))$ is continuous under the small perturbation of $M$ and reduced to the situation of space $M,g,f$ with very nice regularity. The second part is to embed $M$ to a big Euclidean space $\mathbb{R}^N$ as a subspace, and the embedding stays the length of geodesic. Locally, this is determined by a group of PDEs: $u_i(x)u_j(x)=g_{ij}(x)$, at least in the cut locus. But there should be some critical points, and I do not know how to deal with them. The third, i.e. the last step, is to calculate $d_{G-H}(S_n,S_m)$ in the very same deformation space $M,f,g.
@Mark Sapir, Appreciate for help! I am reading the article you point out, it seems this article mainly focuses on investigating the Gromov-Hausdorff limit space of a sequence of hyperbolic groups equipped with modified G-H metric defined in 2.A with some special condition to ensure the limit space exists and take a sequences coverage 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 calculates some Gromov-Hausdorff distance of two different spaces, may you point out it? Appreciate again!
@MarianoSuárez-Álvarez, Corrected, thanks.
You are right, in fact if we use the canonical embed, then we can get $d_{G-H}(S_n,S_m) \leq 2$ by another equivalent definition of GH distance. I confuse the geometry picture of the pairs $T_n,S_n$ with the pairs $S_n,S_m$, for $S_n,S_m$ case, I think the correct conjecture will be $d_{G-H}(S_n,S_m) \sim rac{m-n}{m}, 0 \leq n \leq m, m,n o \infty$.
Clearly from standard embeddings we get $d_{GH}(S_n,S_m) \le \sqrt{2}$ for all $n,m \ge 0$. Would it be reasonable to simply conjecture that it’s an equality whenever $n
eq m$?
Yeah, you are right, $d_{G-H}(S_n,S_m) \leq \sqrt{2}$ for all $n,m \geq 0$. 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 $f:X o Y$ from low-dimensions space $X$ to high-dimension space $Y$ stay some affine structure of the low-dimension space $X$. 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 $S_n,S_m$, if $n,m$ is very near to each other, then the two space should be more near, and there is a canonical embed $S_0 \subset S_1 \subset S_2 \ldots \subset S_n \subset \ldots$. So it is natural to conjecture if $m,n$ is very near then the distance $d_{G-H}(S_n,S_m)$ 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 $d_{G-H}(S_n,S_m)=\inf_{M,g,f}(d_M(f(S_n),g(S_m)))$ then the difficulty is the deformation space of $M,g,f$ is too large. So the first step is to establish a regular lemma, to prove the function $d_M(f(S_n),g(S_m))$ is continuous under the small perturbation of $M$ and reduced to the situation of space $M,g,f$ with very nice regularity.
The second part is to embed $M$ to a big Euclidean space $\mathbb{R}^N$ as a subspace, and the embedding stays the length of geodesic. Locally, this is determined by a group of PDEs: $u_i(x)u_j(x)=g_{ij}(x)$, at least in the cut locus. But there should be some critical points, and I do not know how to deal with them. The third, i.e. the last step, is to calculate $d_{G-H}(S_n,S_m)$ in the very same deformation space $M,f,g.
I need to come back to explain why we expect the Gromov-Hausdorff distance $d_{G-H}(S_d,S_m), 0 \leq n \leq m$ should be much smaller than $\sqrt 2$ when $ rac{n}{m}$ is small.
Let’s 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 endowed with metric. This can be viewed as a complete graph equipped metric, i.e. $M={(G,d_G)}$. So there is also some space very like $S_n, S_m$ in the Euclidean space. Let’s remark them as $G_{S_n}, G_{S_m}$.
Observe that $(G,d_G)\in M$ then $(G,\hat d_{G})\in M$, $\hat d_{G}$ is a scaling of $d_G$ so it is natural to consider a cut off of $M$, called $M_{\lambda}$ which is just a subset of $M$ and satisfied if $(G,d_G)\in M_{\lambda}$, then $\inf_{x
eq y}d_{G}(x,y) \geq d$.
Now, in the space $M_{\lambda}$ let us consider a Distance distribution: $\mu_G((a,b))= rac{# {x,y \in G | a < d_G(x,y) < b}}{#G imes G}$. Then this distribution will give us some information of the distance of the two different set $G_1, G_2$ in $M_{\lambda}$.
And obviously we will see that if $n,m$ is close, then the distribution of fuzzy approximation $G_{S_n}, G_{S_m}$ is near, and the reverse is also true. I think this can explain why the conjecture $d_{G-H}(S_n,S_m) = O( rac{m-n}{m})$ may be right.
By the way, it is a very good exercise to prove $d_{G-H}(S_n,S_0)=1, orall n \in \mathbb{N}^*$.