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 R^{2} but i couldn't really do it .

Could someone please help me out with it ?

Re: Distance and Diameter in a metric space

Quote:

Originally Posted by

**mrmaaza123** 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 R^{2} but i couldn't really do it .

Could someone please help me out with it ?

Just generalized you other example.

Let $\displaystyle A=\{(x,y):x=4\},~B=\{(x,y):\sqrt{x^2+y^2}<1\}\cup A,~\&~x=(1,0)$