FractionalGeometry-Chap2

# A metric d on the set x is a function d x x 0

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Unformatted text preview: ircles, the distance should be positive for r &gt; 0 and should go to 0 as r → 0. How can we do this? 35 2.2. THE HAUSDORFF METRIC The general notion of distance between elements of a set X – think of X as the set of points in the plane, the set of curves in the plane, the set of compact sets in the plane or in space – is captured by the general deﬁnition of a metric. A metric d on the set X is a function d : X × X → [0, ∞) possessing three properties. For all x, y, z ∈ X , (i) d(x, y ) ≥ 0, and d(x, y ) = 0 if and only if x = y (positive-deﬁniteness), (ii) d(x, y ) = d(y, x) (symmetry), and (iii) d(x, y ) ≥ d(x, z ) + d(z, y ) (triangle inequality). The Hausdorﬀ metric is a metric on X = K(R2 ), the set of compact subsets of the plane. Deﬁning the Hausdorﬀ metric requires using the noton of ǫ-thickening. For any compact set A and any number ǫ ≥ 0, the ǫ-thickening of A is Aǫ = {(x, y ) ∈ R2 : d((x, y ), (x′ , y ′ )) ≤ ǫ for some point (x′ , y ′ ) ∈ A} where d is t...
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