Hausdorff metric

English

Hausdorff metric (plural Hausdorff metrics)

(mathematical analysis) In the abstract metric space of all compact subsets of $\mathbb {R} ^{n}$ $\mathbb {R} ^{n}$ , given a pair of compact sets A and B, the Hausdorff metric is $h(A,B)={\mbox{max}}\{\rho (A,B),\rho (B,A)\}$ $h(A,B)={\mbox{max}}\{\rho (A,B),\rho (B,A)\}$ where $\rho (A,B)=\sup _{a\in A}\inf _{b\in B}\,d(a,b)$ $\rho (A,B)=\sup _{a\in A}\inf _{b\in B}\,d(a,b)$ , where d is the Euclidean metric in $\mathbb {R} ^{n}$ $\mathbb {R} ^{n}$ .
Synonyms: Hausdorff distance, Pompeiu-Hausdorff distance
- 2014, James F. Peters, Topology of Digital Images: Visual Pattern Discovery in Proximity Spaces, Springer Science & Business Media, →ISBN, page 195:
  The Hausdorff metric topology was first discovered by D. Pompeiu and by F. Hausdorff in 1927. This topology has been used in a number of areas such as function space topologies, approximation theory, and optimisation (see, e.g., 22).