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Unformatted text preview: S table of PrxsSol 2.2.2. f
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2/3 Figure 2.21: The ﬁrst three stages if the solution to PrxsProb. 2.2.2
Exercises
Prob 2.2.1 (a) Find the Hausdorﬀ distance between the unit circle and the
square with vertices (1, 1), (−1, 1), (−1, −1), and (1, −1).
(b) Find the √
Hausdorﬀ √
distance between the unit circle and the√
square with ver√
√
√
√
√
tices ( 2/2, 2/2), (− 2/2, 2/2), (− 2/2, − 2/2), and (− 2/2, − 2/2).
Prob 2.2.2 Is there a compact set A for which (Aδ1 )δ2 = Aδ1 +δ2 ? Is (Aδ1 )δ2 =
Aδ1 +δ2 true for all compact sets A?
Prob 2.2.3 Complete the proof of Prop. 2.2.1 by showing the Hausdorﬀ distance is symmetric. 2.2. THE HAUSDORFF METRIC 43 Prob 2.2.4 Denote by C the Cantor middle thirds set, the subset of R generated
by the IFS given by T1 (x) = x/3 and T2 (x) = x/3 + 2/3.
(a) Find the smallest ǫ for which Cǫ has no gaps; that is, Cǫ is connected.
(b) Find the smallest ǫ for which Cǫ has exactly one gap; that...
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 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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