# $P,Q$ is any two subset of $(X,d)$ with $P\subset Q$, then what is the value of $\sup\limits_{x\in P} \inf\limits_{y\in Q} d(x,y)$?

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 $$\sup\limits_{x\in P} \inf\limits_{y\in Q} d(x,y)=0$$?

• Small nit-pick: it's $d_H:\mathcal P(X)^2\to\mathcal [0,\infty]$ – Arthur Jul 23 '19 at 12:19

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 $$x$$.
• @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 $Y$." – Martin Sleziak Aug 8 '19 at 9:23