PomeDefinition 15.19.19.0EB9DF723176 For nonempty subsets $E,F$ of a metric space $(W,d_ W)$, their Hausdorff distance is
\[ d_{\mathrm H}^ W(E,F)=\max \left\{ \sup _{x\in E}\inf _{y\in F}d_ W(x,y), \sup _{y\in F}\inf _{x\in E}d_ W(x,y)\right\} . \]
For nonempty bounded metric spaces $E,F$, put
\[ d_{\mathrm{GH}}(E,F) =\inf _{W,\iota ,\jmath }d_{\mathrm H}^ W(\iota (E),\jmath (F)), \]
where $\iota ,\jmath $ are isometric embeddings into a common metric space $W$. A correspondence is a subset $R\subset E\times F$ whose two coordinate projections are surjective. Its distortion is
\[ \operatorname {dis}R =\sup _{(x,y),(x',y')\in R} |d_ E(x,x')-d_ F(y,y')|. \]
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