Hausdorff metric

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This metric arises in normed spaces, giving a measure of distance between two (point) sets, LaTeX: S and LaTeX: T. Given a norm, let LaTeX: d be the distance function between a point and a set:

LaTeX:  d(x, T) = \inf \{\|x-y\|: y \in T\} and LaTeX:  d(S, y) = \inf \{\|x-y\|: x \in S\}.


D(S, T) = \sup \{d(x, T): x \in S\} and LaTeX:  D(T, S) = \sup \{d(S, y): y \in T\}.

Then, LaTeX: h(S, T) = \max \{D(S, T), D(T, S)\} is the Hausdorff metric. In words, this says that for each LaTeX: x \in S, let LaTeX: y(x) be its closest point in LaTeX: T. Then, maximize this distance among all LaTeX: x.

For example, let LaTeX: S be the interval, LaTeX: [-1, 1], and let LaTeX: T be the interval, LaTeX: [0, 3]. The Hausdorff distance between these two intervals is:

h(S, T) = \max \{D(S, T), D(T, S)\} = \max\{1, 2\} = 2.

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