1.8 The Hausdorff metric
The set of convex bodies can be made into a metric space in several geometrically reasonable ways. The Hausdorff metric is particularly convenient and applicable. The natural domain for this metric is the set Cn of nonempty compact subsets of We shall first treat the Hausdorff metric on this more general class of sets; this will involve no extra effort and is useful in several respects.
The Hausdorff distance of the sets K, L ∈ Cn is defined by
or, equivalently, by
Then δ is a metric on Cn, the Hausdorff ...