HAUSDORFF METRIC

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

d(x, T) = inf(||x-y||: y in T) and d(S, y) = inf(||x-y||: x in S). Define

D(S, T) = sup(d(x, T): x in S) and D(T, S) = sup(d(S, y): y in T). Then, h(S, T) = max(D(S, T), D(T, S)) is the Hausdorff metric. In words, this says that for each x in S, let y(x) be its closest point in T. Then, maximize this distance among all x. For example, let S be the interval, [-1, 1], and let T be the interval, [0, 3]. The Hausdorff distance between these two intervals is:

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