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I want to calculate the Gromov Hausdorff distance between the unit sphere $S^2$ and the unit interval $[0,1]$.

From what I understand of this example, I need to make a gluing of $S^2$ and $[0,1]$ and find an admissible metric. I don't know how to do this, and also I don't know what an admissible metric is.

How can I calculate $d_{GH}(S^2,[0,1])$?

Any help would be appreciated.

Thanks.

  • I forgot to mention, but I am considering the euclidean metrics on $S^2$ and $[0,1]$ –  Sep 05 '20 at 17:39
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    Why would you want to do this? Take a look here especially the quote of Burago. – Moishe Kohan Sep 05 '20 at 19:08
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    I'm studying the Gromov Hausdorff distance and would like to calculate it on some examples. This one was chosen by myself. After reading that quote of Burago, I'm realizing choosing an example is more difficult than I thought. –  Sep 06 '20 at 00:39
  • @MoisheKohan, do you happen to know what is an "admissible metric"? I can't find that anywhere. –  Sep 06 '20 at 00:46
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    I've never heard of this terminology. – Moishe Kohan Sep 06 '20 at 01:08
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    @user1618 Given two metric spaces $X, Y$, an admissible metric is a metric on $X \amalg Y$ (the disjoint union of $X$ and $Y$) which restricts to the respective metrics on both $X$ and $Y$. It can be shown that the best Hausdorff distance between $X$ and $Y$ under an admissible metric, is equal to the Gromov Hausdorff distance. The argument in the second part of the attached post seems wrong to me. I attempted an answer, but had to delete due to an incorrect calculation. Lastly, I believe we cannot get an exact answer to your question using the current tools. – Sarvesh Ravichandran Iyer Feb 21 '21 at 14:27
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    While I don't know about exactness, I read on Math Overflow that $d_{GH} (S^2 , [-1,1]) = 1$, and apparently it is a nice exercise. If we can get hold of $d_{GH}([0,1],[-1,1])$ then we can use the triangle inequality to get an upper bound. I suspect the answer to the second question should be easy. For a lower bound, we can use the diameter inequality , since $diam(S^2) = 2$ and $diam([0,1]) = 1$ the lower bound is $\frac 12$. At least this helps bound the distance. What it is exactly, will be difficult to know. – Sarvesh Ravichandran Iyer Feb 23 '21 at 03:21
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    Example in mathoverflow is about an induced metric from $\mathbb{R}^3$, as you said. Hence the diameter is $2$. And OP may want about the intrinsic metric on $\mathbb{S}^2$, whose diameter is $\pi$. – HK Lee Feb 23 '21 at 09:04

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