Hausdorff metric

From hyperspacewiki
Jump to: navigation, search

Let $(X,d)$ be a metric space. Let $$N_d(r,A) = \{x \in X \colon d(x,A) < r\}.$$ Let $A,B \in \mathrm{CL}(X)$, the hyperspace of closed subsets of $X$. The Hausdorff metric induced by $d$ is the function $$H_d(A,B) = \inf \{ r > 0 \colon A \subset N_d(r,B) \mathrm{ \hspace{2pt} and \hspace{2pt}} B \subset N_d(r,A)\}.$$ Equivalently, the Hausdorff metric can be defined as $$H_d(A,B) = \max \left\{\sup_{a \in A} \inf_{b \in B} d(a,b), \sup_{b\in B}\inf_{a \in A} d(a,b) \right\}.$$

Theorem

The Hausdorff metric is a metric on CL(x)

References

Illanes, Alejandro ; Nadler, Sam B., Jr. Hyperspaces. Fundamentals and recent advances. Monographs and Textbooks in Pure and Applied Mathematics, 216. Marcel Dekker, Inc., New York, 1999. xx+512 pp. ISBN: 0-8247-1982-4