Let $(X,d_1)$ be a metric space. Assume there are $x,y \in X$, such that $$d_1(x,z)\leq d_1(y,z) \quad \forall z \in X\setminus \lbrace x,y\rbrace. \quad (*) $$
I am trying to show that if one defines another metric $d_2$ on X, this property is preserved, i.e.
$$(*) \implies d_2(x,z)\leq d_2(y,z) \quad \forall z \in X\setminus \lbrace x,y\rbrace $$
I am not sure if it is actually possible to prove this in general. I guess one should rather focus on properties, that $d_2$ has to fullfil in order to preserve $(*)$. I tried it with topollogically equivalent metrics but so far i couldn't come up with a proof.
Any suggestion towards proving this hypothesis would be helpful.