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Math Help - Distance and Diameter in a metric space

  1. #1
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    Distance and Diameter in a metric space

    Find a metric space X, an element "x" of X and non empty subsets A and B of X where A is a subset of B, such that dist(x,A) > dist(x,B) + diam(B/A) ?

    I managed to find one in R.
    I took X = (R,d)
    A = {4}
    B = (0,1) U {4}
    x =1
    dist(x,A) = 3 ,dist(x,B) = 0
    diam(B/A) = 1
    Hence the inequality is satisfied.
    I was trying to construct such an example in R2 but i couldn't really do it .
    Could someone please help me out with it ?
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    Re: Distance and Diameter in a metric space

    Quote Originally Posted by mrmaaza123 View Post
    Find a metric space X, an element "x" of X and non empty subsets A and B of X where A is a subset of B, such that dist(x,A) > dist(x,B) + diam(B/A) ?

    I managed to find one in R.
    I took X = (R,d)
    A = {4}
    B = (0,1) U {4}
    x =1
    dist(x,A) = 3 ,dist(x,B) = 0
    diam(B/A) = 1
    Hence the inequality is satisfied.
    I was trying to construct such an example in R2 but i couldn't really do it .
    Could someone please help me out with it ?
    Just generalized you other example.
    Let A=\{(x,y):x=4\},~B=\{(x,y):\sqrt{x^2+y^2}<1\}\cup A,~\&~x=(1,0)
    Thanks from mrmaaza123
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