Hi!, I have problems with this Proposition
PROPOSITION
Let and metric space. For any we have
, where ,
,
I have problems with proof the inequalities
It is a geometric property of triangles that the sum of the lengths of two sides is always greater than the third. So, for 2 dimensions, the inequality follows immediately from the fact that d(x,y) is the length of the hypotenuse while d'(x,y) is either that exact same length or it is the sum of the lengths of the other two sides of the triangle. For higher dimensions, you would need a generalization of that fact.
http://www.cs.bc.edu/~alvarez/NDPyt.pdf
This gives an idea of how to start.