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Given a metric space $(X,d)$, take $A,B\subset X$. I am searching for a notion of distance between two sets which expresses the "maximum" distance between them in a coherent way.

I'm going to explain.

Consider the following two parabolas in $\Bbb R^2$: $$ A:=\{(x,x^2),\;x\in\Bbb R\} $$ and $$ B:=\{(x,-x^2),\;x\in\Bbb R\}\;. $$

Now if I consider the usual notion of distance between two sets (call it $d_1$), we have $$ d_1(A,B):=\inf_{a\in A,b\in B}d(a,b)=0. $$ This is not satisfactory to me, because it expresses the closest point of the two curves: we have no information, no control on the rest of the curves.


Of course a definition taking into account all the curve could be $$ d_2(A,B):=\sup_{a\in A,b\in B}d(a,b) $$ but it is clearly bad posed: we can take any point in each curve, without relation one with the other. This doesn't express anything: take for example the following two lines in $\Bbb R^2$: $$ C:=\{y=\varepsilon/2\}\\ D:=\{y=-\varepsilon/2\}\\ $$ for some small $\varepsilon>0$. By looking at them one feels they are "very close", but with this last definition, one has $d_2(C,D)=\infty$.

So this is clearly a bad definition.


Let us come back to the curves $A$ and $B$: for any point in one of them, say $a\in A$, we can consider the distance between it and the other curve, that is $\inf_{b\in B}d(a,b)$; but in order to avoid the arbitrary choice of the point we should do this for every point and taking the supremum of all such distances to finally express a uniform closeness of two sets $A$ and $B$. Namely: $$ d_\infty(A,B):=\sup_{x\in X}\left[\inf_{y\in Y}d(x,y)\right]\;. $$ With this definition we would have for example $$ d_{\infty}(A,B)=\infty $$ and $$ d_{\infty}(C,D)=\varepsilon $$ which looks coherent.

I wonder if such a definition is described somewhere, hopefully with a systematic presentation, properties and so on. Thank you.

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I don't know what your criteria are for a good answer, but I'll mention the Hausdorff distance, mainly for the reason that it actually has wide use and applicability in geometry.

First, for any $A \subset X$ and $r > 0$ let $$N_r(A) = \{y \in X \mid \exists x \in A, d(x,y) < r\} $$ The Hausdorff distance between $A$ and $B$ is defined to be $$d_H(A,B) = \inf\{r \mid B \subset N_r(A) \quad\text{and}\quad A \subset N_r(B)\} $$ In your two examples, $d_H(A,B)=+\infty$ whereas $d_H(C,D)=\epsilon$.

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