Gromov-Hausdorff distance ([Wikipedia](https://en.wikipedia.org/wiki/Gromov%E2%80%93Hausdorff_convergence)) between two compact manifolds measures how far away the manifolds are from being isometric.
In many cases it is possible to do coarse estimates and conclude that a sequence of manifolds converges or diverges.


How does one usually go about calculating GH distance precisely?


Example: Take two spheres of different radii $r$ and $R$ with intrinsic (i.e the distance between two points is the length of the arc of a great circle that connects them) metrics obtained from standard embeddings into $\Bbb R^n$. What is the GH distance between them?



What are the tricks for computing/estimating Gromov-Hausdorff distance?
I misinterpreted the question at first, sorry. Here's my new answer:


**First, an answer to the wrong question**


For two $(n-1)$ dimensional spheres of radii r and R with the metrics induced from embedding in $\mathbb{R}^n$ (note, this is the "chord metric" not the "round metric" as Zarathustra desired), the Gromov-Hausdorff distance is $|r-R|$. We can achieve this as an upper bound by embedding the two spheres in a concentric fashion, and it's seen to be sharp by the inequality $d\_{GH}(X,Y)\geq \frac{1}{2}|\operatorname{diam}(X)-\operatorname{diam}(Y)|$.


See e.g. [Burago Burago and Ivanov, ex. 7.3.14](https://books.google.com/books?id=afnlx8sHmQIC&pg=PA255&dq=gromov+hausdorff+7.3.14&ei=3h7aS8ekIqeeygSR0OGtCQ&cd=1#v=onepage&q=gromov%20hausdorff%207.3.14&f=false) which is a good source in general.


**Now an answer to the correct question**


The answer is $\frac{\pi}{2}|R-r|$ for spheres with the round metric, as Anton more or less suggested. This follows easily from the discussion following [defn. 7.3.17 in BBI](https://books.google.com/books?id=afnlx8sHmQIC&pg=PA256&lpg=PA258&dq=burago+burago+ivanov+correspondence&source=bl&ots=JvJ19H2X98&sig=VVQSTNM35zJq_hecDlDC3Jpd1pk&hl=en&ei=6lHaS-u2KJPy9QS96ORF&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBIQ6AEwAQ#v=onepage&q&f=false).


Out of laziness, I've written out some of the details here. A "correspondence" of metric spaces $X$ and $Y$ is defined to be a subset $\mathcal{R}$ of $X\times Y$ such that for every point $x\in X$ there is at least one point $(x,z)\in\mathcal{R}$ and for every $y\in Y$ there is at least one point $(w,y)\in\mathcal{R}$. From this one can prove Theorem 7.3.25 which states


$d\_{GH}(X,Y)=\frac{1}{2}\inf\_{\mathcal{R}}dis\mathcal{R}$


where the infimum is taken over all correspondences $\mathcal{R}$ and $dis\mathcal{R}$ is the distortion of $\mathcal{R}$, defined to be $\sup\{|d\_X(x,x')-d\_Y(y,y')|:(x,y),(x',y')\in\mathcal{R}\}$.


Take $\mathcal{R}$ to be the correspondence consisting of pairs $(x,y)$ with $x\in S^2\_{r}$ and $y\in S^2\_{R}$ if $x$ and $y$ lie on the same ray through the origin when the two spheres are embedded in $\mathbb{R}^3$. The distortion of this correspondence is $\pi|R-r|$ by taking $x$ and $x'$ to be antipodal points on one of the spheres. This gives an upper bound for $d\_{GH}$ of $\frac{\pi}{2}|R-r|$, and this is sharp again by the inequality above.


**A possibly useful reference for G-H distance for subsets of Euclidean space in general**


For the rest of your question, you might find interesting [a paper by Facundo Mémoli](https://people.math.osu.edu/memoli.2/papers/dgh-euclidean.pdf) which discusses the case when X and Y are subsets of Euclidean space. See also slides [here](https://people.math.osu.edu/memoli.2/talks/dgh-euclidean-cvpr.pdf).


