Gromov-Hausdorff distance – Relationship of relationship

Introduction

the statement go isometry inequality is very simple:

$\Omega\subset R^n$ , iff $\Omega$ is a ball, $\frac{Vol(\Omega)}{Surf(\Omega)}$ arrive a minimum .

This is a classical problem in variation theory. The difficult is divide into two parts. The first is to create a “flow” which descrement the energy and the “flow” is compatible with the feature of a ball, i.e. every set under the flow will tend to like a “ball”. The second one is to proof there exist a unit in the space $surf(\Omega)=constant>0$ make the Energy $E(\Omega)=Vol(\Omega)$ arrive a minimum.

Combine this two property we can consult that ball is the set and definitely the only set make the $\frac{Vol(\Omega)}{Surf(\Omega)}$ arrive the minimum.

first difficulties

The energy $E(\Omega)$ is scaling invariance. The first difficult could divide into two part:

Restrict to convex set

the first is to deform a set into a convex set and proof this process would not lower $\frac{Vol(\Omega)}{Surf(\Omega)}$ . This could been a little subtle. and the way I image could make sense is just like the following transform:

QQ20171116-151122@2x

but this process is harder in higher dimension, for example:

0D1371F0-19C9-480B-8804-EC2281B6A0C8

Convex set to a ball

Steiner symmetric process.

Affine transform

Minkowski–Steiner formula
In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the “derivative” of enclosed volume in an appropriate sense.

The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after Hermann Minkowski and Jakob Steiner.

Statement of the Minkowski-Steiner formula

Let $n \geq 2$ , and let $A \subsetneq \mathbb{R}^{n}$ be a compact set. Let $\mu (A)$ denote the [[Lebesgue measure]] (volume) of $A$ . Define the quantity $\lambda (\partial A)$ by the ”’Minkowski–Steiner formula”’:

$\lambda (\partial A) := \liminf_{\delta \to 0} \frac{\mu \left( A + \overline{B_{\delta}} \right) - \mu (A)}{\delta}$

where:

$\overline{B_{\delta}} := \left\{ x = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n} \left| | x | := \sqrt{x_{1}^{2} + \dots + x_{n}^{2}} \leq \delta \right. \right\}$

denotes the [[closed ball]] of [[radius]] $\delta > 0$ , and:

$A + \overline{B_{\delta}} := \left\{ a + b \in \mathbb{R}^{n} \left| a \in A, b \in \overline{B_{\delta}} \right. \right\}$

is the [[Minkowski sum]] of $latexA$ and $\overline{B_{\delta}}$ , so that:

$A + \overline{B_{\delta}} = \left\{ x \in \mathbb{R}^{n} | |x - a| \leq \delta \mbox{ for some } a \in A \right\}$ .

Surface measure

For “sufficiently regular” sets $A$ , the quantity $\lambda (\partial A)$ does indeed correspond with the $(n - 1)$ -dimensional measure of the [[boundary (topology)|boundary]] $\partial A$ of $A$ . See Federer (1969) for a full treatment of this problem.

Convex sets

When the set $A$ is a [[convex set]], the [[limit inferior|lim-inf]] above is a true [[Limit of a sequence|limit]], and one can show that

: $\mu \left( A + \overline{B_{\delta}} \right) = \mu (A) + \lambda (\partial A) \delta + \sum_{i = 2}^{n - 1} \lambda_{i} (A) \delta^{i} + \omega_{n} \delta^{n}$ ,

where the $\lambda_{i}$ are some [[continuous function]]s of $A<$ (see [[quermassintegral]]s) and $\omega_{n}$ denotes the measure (volume) of the [[unit ball]] in $\mathbb{R}^{n}$ :

: $\omega_{n} = \frac{2 \pi^{n / 2}}{n \Gamma (n / 2)}$ ,

where $\Gamma$ denotes the [[Gamma function]].

==Example: volume and surface area of a ball==

Taking $A = \overline{B_{R}}$ gives the following well-known formula for the surface area of the [[sphere]] of radius $R$ , $S_{R} := \partial B_{R}$ :

: $\lambda (S_{R}) = \lim_{\delta \to 0} \frac{\mu \left( \overline{B_{R}} + \overline{B_{\delta}} \right) - \mu \left( \overline{B_{R}} \right)}{\delta}$
:: $= \lim_{\delta \to 0} \frac{[ (R + \delta)^{n} - R^{n} ] \omega_{n}}{\delta}$
:: $= n R^{n - 1} \omega_{n}$ ,

where $\omega_{n}$ is as above.

something with more details

The second difficulties

To establish a continue property of the Energy functional $E(\Omega)= \frac{Vol(\Omega)}{Surf(\Omega)}$ .

the continuous property is consider with all open set $\Omega$ with Gromov-Hausdorff metric $d(\Omega_1,\Omega_2)= \inf_{metric\ d on \Omega_1 \cup \Omega_2}\sup_{x_1\in \Omega_1, x_2\in \Omega_2}d(x_1,x_2)$ .

We need to proof the continuous of $E(\Omega)$ with the Gromov-hausdorff metric on the space consist with convex open sets.

To remark,we need to observe that polygon approximation is just corresponding to the $\delta-seperate$ points approximation in Gromov-hausdorff distance. and definitely carefully refinement of this kind of approximation could lead to the result of continuous of the energy $E(\Omega)$ on convex set.

A second remark, we definitely need a definition of the $surf(\Omega)$ it could be achieve with open convex set $\Omega$ by a outer and inter approximation by polygon and the error term estimate.

Further remark, isoperimetric inequality is a general phenomenon.

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.

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 $S_0\subset S_1 \subset S_2 ....\subset S_n \subset ...$ . 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 difficult 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 continues under the small perbutation 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 euclid space $R^N$ as subspace, and the embedding stay the length of geodesic.locally this is determine by a group of pde: $u_i(x)u_j(x)=g_{ij}(x)$ ,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 $d_{G-H}(S_n,S_m)$ in the very some deformation space $M,f,g$ .

@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 $\pi$ when endowed with geodesic distance) and hence the GH distance between any two spheres is $\le 4$ .

H:
73
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 thick the seems correct conjecture will be $d_{G-H}(S_n,S_m)\sim \frac{m-n}{m},0\leq n\leq m,m,n\to \infty$ .

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

H:
73
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 $f:X\to Y$ 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 $S_0\subset S_1 \subset S_2 ....\subset S_n \subset ...$ . 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 difficult 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 continues under the small perbutation 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 euclid space $R^N$ as subspace, and the embedding stay the length of geodesic.locally this is determine by a group of pde: $u_i(x)u_j(x)=g_{ij}(x)$ ,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 $d_{G-H}(S_n,S_m)$ in the very some deformation space $M,f,g$ .
I need come back to explain why we expect the groom-hausdorff distance $d_{G-H}(S_d,S_m),0\leq n\leq m$ should be much small than $\sqrt 2$ when $frac{n}{m}$ 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. $M=\{(G,d_G)\}$ . So there is also some space very like $S_n$ ,$S_m$ in the Euclid space, Let remark them as $G_{S_n},G_{S_m}$ .
, oberseve 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 $\inf_{x\neq y}d_{G}(x,y)\geq d$ .
Now,in the space $M_{\lambda}$ let us consider a Distance distribution: $\mu_G((a,b))=\frac{\#\{x,y\in G|a<d_G(x,y)<b\}}{\#G\times G}$ . Then this distribution will give us some information of the distance of the two different set $G_1,G_2$ in $M_{\lambda}$ .
(removed)
Now,in the space $M_{\lambda}$ let us consider a Distance distribution: $\mu_G((a,b))=\frac{\#\{x,y\in G|a<d_G(x,y)<b\}}{\#G\times G}$ . Then this distribution will give us some information of the distance of the two different set $G_1,G_2$ 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 $d_{G-H}(S_n,S_m) =O(\frac{m-n}{m})$ may be right.