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Let $(X,d)$ be a metric space. Then the function $d_H:\mathcal P(X)\times\mathcal P(X)\to\mathcal [0,\infty]$ defined by $$d_H(A,B)= \max \{ \sup\limits_{x\in A} \inf\limits_{y\in B} d(x,y), \ \sup\limits_{y\in B} \inf\limits_{x\in A} d(x,y) \}$$ for all $A,B\subset X$. This $d_H$ is called Hausdorff distance.

Now I want to check $d_H (A,B)$ for some particular case.

Consider the metric space $(\mathbb R^2,d)$ with euclidean metric $d$. Let $B=\{(x,y):x^2+y^2\leq16 \}$ and $A=\{(x,y):(x+2)^2+y^2\leq1 \}$ two subsets of $\mathbb R^2$. Here $A\subset B$. Then I have checked that first part of the definition of $d_H$ is $0$ and second part is $5$. Thus $d_H (A,B)$ is equal to $5$, being the maximum of $0,5$. Am I wrong?

I am confused because of this figure in Wikipedia.

My question is if $P,Q$ is any two subset of $(X,d)$ with $P\subset Q$, then is it true that $\sup\limits_{x\in P} \inf\limits_{y\in Q} d(x,y)=0$?

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    $\begingroup$ Small nit-pick: it's $d_H:\mathcal P(X)^2\to\mathcal [0,\infty]$ $\endgroup$ – Arthur Jul 23 '19 at 12:19
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First question: I think your answer is correct. $d_H(A,B)=5$.

Second question. YES, again. If $x \in P$ then $x \in Q$ so $\inf_{y \in Q} d(x,y)\leq d(x,x)=0$. Take sup over $x$.

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  • $\begingroup$ Then I believe that the above mentioned figure from Wikipedia is actually wrong. @Murthy sir, have you seen it? $\endgroup$ – BijanDatta Jul 23 '19 at 16:46
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    $\begingroup$ @BijanDatta Yes, the figure is wrong. $\endgroup$ – Kavi Rama Murthy Jul 23 '19 at 23:16
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    $\begingroup$ @BijanDatta I don't really follow what you mean by saying that the figure is "wrong". I don't see anywhere in that Wikipedia article the claim that $d_H$ is zero in that picture. Moreover, the picture displays distance between two sets neither of which is a subset of the other one. (The current revision says in the caption of the figure: "Components of the calculation of the Hausdorff distance between the green line $X$ and the blue line $Y$." $\endgroup$ – Martin Sleziak Aug 8 '19 at 9:23
  • $\begingroup$ Actually it seems from the picture that the two sets are in the situation that one is contained in another (though it is not written in Wikipedia for this picture). Then the length of one line(among blue and green) should be zero. So the picture is wrong in that sense what I've meant here. I know this was a stupid concept to claim. $\endgroup$ – BijanDatta Aug 8 '19 at 9:55

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