# Distance and Diameter in a metric space

• Sep 21st 2013, 10:38 PM
mrmaaza123
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 .
• Sep 22nd 2013, 02:58 AM
Plato
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 R2 but i couldn't really do it .
Let $\displaystyle A=\{(x,y):x=4\},~B=\{(x,y):\sqrt{x^2+y^2}<1\}\cup A,~\&~x=(1,0)$