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Diameters of Sets

  1. Oct 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Find a condition on a metric space [itex](X,d)[/itex] that ensures that there exist subsets [itex]A,B[/itex] of [itex]X[/itex] with [itex]A\subset B[/itex] such that [itex]diam(A)=diam(B)[/itex].

    2. Relevant equations
    [itex]diam(A)=\sup\{d(r,s):r,s\in A\}[/itex];
    [itex]A\subseteq B\implies diam(A)\leq diam(B)[/itex].

    3. The attempt at a solution
    Well I know examples of where this is true (ie, let [itex]A=(-\infty,5]\cup [-5,\infty)\subset (-\infty,4]\cup [-4,\infty)=B[/itex]). But I don't know which condition allows this to be true. Any help is good. Thank you!
  2. jcsd
  3. Oct 12, 2011 #2
    Bump. Anyone? Any Idea?
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