In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu. Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance someone can be forced to travel by an adversary who chooses a point in one of the two sets, from where they then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set. This distance was first introduced by Hausdorff in his book Grundzüge der Mengenlehre, first published in 1914, although a very close relative appeared in the doctoral thesis of Maurice Fréchet in 1906, in his study of the space of all continuous curves from [ 0 , 1 ] → R 3 {\displaystyle [0,1]\to \mathbb {R} ^{3}} .
Definition
Let ( M , d ) {\textstyle (M,d)} be a metric space. For each pair of non-empty subsets X ⊂ M {\textstyle X\subset M} and Y ⊂ M {\textstyle Y\subset M} , the Hausdorff distance between X {\textstyle X} and Y {\displaystyle Y} is defined as
d H ( X , Y ) := max { sup x ∈ X d ( x , Y ) , sup y ∈ Y d ( X , y ) } , {\displaystyle d_{\mathrm {H} }(X,Y):=\max \left\{\,\sup _{x\in X}d(x,Y),\ \sup _{y\in Y}d(X,y)\,\right\},}
where sup {\textstyle \operatorname {sup} } represents the supremum operator, d ( a , B ) := inf b ∈ B d ( a , b ) {\textstyle d(a,B):=\displaystyle \inf _{b\in B}d(a,b)} quantifies the distance from a point a ∈ X {\textstyle a\in X} to the subset B ⊆ X {\textstyle B\subseteq X} , and inf {\textstyle \operatorname {inf} } is the infimum operator. An equivalent definition is as follows. For each set X ⊂ M , {\displaystyle X\subset M,} let
X ε := ⋃ x ∈ X { z ∈ M ∣ d ( z , x ) < ε } , {\displaystyle X_{\varepsilon }:=\bigcup _{x\in X}\{z\in M\mid d(z,x)<\varepsilon \},}
which is the set of all points within ε {\textstyle \varepsilon } of the set X {\textstyle X} (sometimes called the ε {\textstyle \varepsilon } -fattening of X {\textstyle X} or a generalized ball of radius ε {\textstyle \varepsilon } around X {\textstyle X} ). Then, the Hausdorff distance between X {\textstyle X} and Y {\textstyle Y} is defined as
d H ( X , Y ) := inf { ε ≥ 0 ∣ X ⊆ Y ε and Y ⊆ X ε } . {\displaystyle d_{\mathrm {H} }(X,Y):=\inf\{\varepsilon \geq 0\mid X\subseteq Y_{\varepsilon }{\text{ and }}Y\subseteq X_{\varepsilon }\}.}
Equivalently,
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