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Convex Bodies: The Brunn–Minkowski Theory, Second Edition by Rolf Schneider

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1.8 The Hausdorff metric

The set image 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 image 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, LCn is defined by

    (1.59)

or, equivalently, by

    (1.60)

Then δ is a metric on Cn, the Hausdorff ...

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