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Suppose A {\displaystyle A} and B {\displaystyle B} are two subsets of a metric space ( X , d ) {\displaystyle (X,d)} . The Hausdorff distance between A {\displaystyle A} and B {\displaystyle B} , denoted d H ( A , B ) {\displaystyle d_{H}(A,B)} , is defined as:
d H ( A , B ) = inf { ε ∣ A ∈ B ε ( B ) , B ∈ B ε ( A ) } {\displaystyle d_{H}(A,B)=\inf\{\varepsilon \mid A\in B_{\varepsilon }(B),B\in B_{\varepsilon }(A)\}} .
Here B ε ( M ) {\displaystyle B_{\varepsilon }(M)} denotes the union of balls of radius ε {\displaystyle \varepsilon } about all points in M {\displaystyle M} .
and 
are two subsets of a metric space 
. The Hausdorff distance between and , denoted 
, is defined as:
.
denotes the union of balls of radius 
about all points in 
.