# Notion of “sup-distance” between two sets

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.

## 1 Answer

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 $$d_H(C,D)=\epsilon$$.