1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook