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I want to prove the following statement, where $d_{GH}$ denotes the Gromov-Hausdorff distance:

For $X= S^1 \times [0,1]$ and $Y= [0,1]$ with the intrinsic metrics one has $d_{GH}(X,Y) = \frac{\pi}{2} $

What I found in the literature is the following:

"$\leq$": define an admissible metric on $X \amalg Y$ by $d((\theta,t),s)= \sqrt{ (\frac{\pi}{2})^2+ (t-s)^2} $. This implies $d_{GH}(X,Y) \leq \frac{\pi}{2} $.

For "$\geq$".
Take two antipodal points $p,q$ in $S^1$ and any admissible metric $d$ on $X \amalg Y$. For every $y \in Y$ one has $\pi = d(p,q) \leq d(p,y) + d(y,q) \Rightarrow \max \{d(p,y),d(y,q)\} \geq \pi/2$ This implies "$\geq$".

In particular I do not understand the last part. It is true that for every $y$ one of the points $p$ or $q$ satiesfies $d(p,y) \geq \pi/2$. But there could be another point $y*$ with $d(p,y*) \leq \pi/2$ and one can no longer conclude $d_{GH}(X,Y) \geq \pi/2$.

HK Lee
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asterisk
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  • Does $S^1$ mean unit radius circle with arclength metric? Then the distance between two antipodal points of $S^1$ is $\pi$, not $\pi/2$, so the inequality $\le$ does not hold. Please clarify. –  Jul 27 '14 at 21:49
  • $S^1$ is supposed to be the unit circle with arclength metric. There is a mistake in my argumentation, I will modify it in a minute. – asterisk Jul 28 '14 at 08:16
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    Your objection to the proof is valid. Somehow this argument does not use that the length of the line segment equals the height of the cylinder, and without this assumption the conclusion is wrong. E.g., you can take a very long line segment and wind it pretty densely around the cylinder, giving a small Gromov-Hausdorff distance. Unfortunately, I can't think of a fix right now. – Lukas Geyer Feb 09 '16 at 19:41
  • You don't specify how the metric is defined on the product space $S^1\times[0,1]$. The truth of the claim depends on that. – George Lowther Feb 11 '16 at 20:33
  • @George Lowther The sphere and the intervall carry their standard (intrinsic) metrics and the product space carries the standard product metric. – asterisk Feb 12 '16 at 22:26
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    @asterisk I am not able to procure the book by Xiaochun Rong. Is it possible if you can provide me a link, or a way to get the entire chapter which discusses this topic? I liked the paper here by the way, you can read it. It won't answer your question, but goes into further detail on the calculation of GH distance in general. – Sarvesh Ravichandran Iyer Feb 21 '21 at 16:30

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