Hausdorff distance - Knowlpedia

Definition

For nonempty compact subsets $A,B$ of a metric space $(X,d)$ , their Hausdorff distance is

d_H(A,B)=\max\left\{ \sup_{a\in A}d(a,B),\qquad \sup_{b\in B}d(b,A) \right\},

where $d(x,C)=\inf_{c\in C}d(x,c)$ .

Neighborhood form

Equivalently,

d_H(A,B)=\inf\{\varepsilon>0: A\subseteq B_\varepsilon\text{ and }B\subseteq A_\varepsilon\}.

On the nonempty compact subsets of $X$ , this is a genuine metric. For more general closed unbounded subsets it may be infinite.

Convex bodies

On convex bodies in a finite-dimensional vector space, the topology induced by $d_H$ does not depend on the chosen Euclidean norm. Under a fixed norm it agrees with uniform convergence of support functions on the dual unit sphere.

References

Dmitri Burago, Yuri Burago, and Sergei Ivanov, A Course in Metric Geometry, AMS, 2001. DOI record. Relevant: §7.3.
Rolf Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd ed., Cambridge University Press, 2014. DOI record. Relevant: §1.8.