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Homework Help: Hausdorff Distance Proofs

  1. Sep 13, 2011 #1
    1. The problem statement, all variables and given/known data
    Given a compact set A[itex]\subset[/itex][itex]\Re[/itex][itex]^{n}[/itex] and a point x[itex]\in[/itex][itex]\Re[/itex][itex]^{n}[/itex] define the distance from x to A as the quantity:

    d(x, A)=inf({[itex]\left\|[/itex]x-y[itex]\right\|[/itex]: y[itex]\in[/itex]A})

    Given two compact sets A, B [itex]\subset[/itex][itex]\Re[/itex][itex]^{n}[/itex], define the Hausdorff distance between them to be:

    d(A, B)=max(sup{d(x, B) : x[itex]\in[/itex]A}, sup{d(x, A) : x[itex]\in[/itex]B})

    a. For any compact set A, prove that the function f : [itex]\Re[/itex][itex]^{n}[/itex][itex]\rightarrow[/itex][itex]\Re[/itex] given by: f(x)=d(x, A) is continuous.

    b. For any two compact sets A, B it's true that: d(A, B)<[itex]\infty[/itex]

    c. For any two compact sets A, B it's true that d(A, B)=0 if and only if A=B.




    2. Relevant equations



    3. The attempt at a solution

    I think I handled part a. I'm just completely lost on b and c. Any help?
     
    Last edited: Sep 14, 2011
  2. jcsd
  3. Sep 13, 2011 #2
    The \left\ and \right\ should be norm.
     
  4. Sep 14, 2011 #3
    Edited for a typo.
     
  5. Sep 14, 2011 #4
    For (b), you need to show that

    [tex]\sup_{x\in A}{d(x,B)}[/tex]

    is bounded and equivalently that

    [tex]\sup_{x\in B}{d(x,A)}[/tex]

    is bounded. Now, can you use (a) to show this?
     
  6. Sep 15, 2011 #5
    I got it! I really appreciate the help. I have one more question.

    How do I show the following:

    For any three compact sets: A, B, C we have a triangle inequality: d(A, C)[itex]\leq[/itex] d(A,B) + d(B, C)
     
  7. Sep 15, 2011 #6
    Let

    [tex]a(A,B)=\sup_{a\in A}{d(a,B)}[/tex]

    (then [itex]d(A,B)=\max{a(A,B),a(B,A)}[/itex]).

    Try to show first that

    [tex]a(A,B)\leq a(A,C)+a(C,B)[/tex]

    Thus, show the triangle inequality first for a.
     
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