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Distance between two sets A and B in Rn

  1. Nov 29, 2009 #1
    1. The problem statement, all variables and given/known data

    A is a sequentially compact set, and B is a point ⃗v in Rn−A.
    define the distance between A and B as dist( ⃗u0 , ⃗v), where you showed
    that ⃗u0∈ A exists such that dist( ⃗u0 , ⃗v)≤dist(⃗u, ⃗v) for all ⃗u in A.
    a) Use this example to state a definition of the distance between two sets in general,
    giving the largest class of sets for which your definition works.
    b) Prove that your definition is well defined on the class of sets for which you say it
    applies in a), and give counterexamples showing you can’t weaken the conditions in your
    definition.
    c) think of another definition of distance which applies to any two sets in Rn . Check that this second definition gives a reasonable answer for the distance between an open ball and its boundary. If you restrict this second definition to the class of sets considered in a) you should get the same answer as a).


    2. Relevant equations



    3. The attempt at a solution

    I don't have any clue of writing the definition and the proof. What does the largest class of sets mean?
     
  2. jcsd
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