Hausdorff distance

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 you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you 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 . Let X and Y be two non-empty subsets of a metric space . We define their Hausdorff distance by where sup represents the supremum, inf the infimum, and where quantifies the distance from a point to the subset . Equivalently, where that is, the set of all points within of the set (sometimes called the -fattening of or a generalized ball of radius around ). Equivalently, that is, , where is the smallest distance from the point to the set . It is not true for arbitrary subsets that implies For instance, consider the metric space of the real numbers with the usual metric induced by the absolute value, Take Then . However because , but . But it is true that and ; in particular it is true if are closed. In general, may be infinite. If both X and Y are bounded, then is guaranteed to be finite. if and only if X and Y have the same closure. For every point x of M and any non-empty sets Y, Z of M: d(x,Y) ≤ d(x,Z) + dH(Y,Z), where d(x,Y) is the distance between the point x and the closest point in the set Y.