Let $X$ be a compact metric space. Assume that $X$ has more than $2$ points (or even better, that $X$ is connected with more than 1 point). Given a metric $d$ on $X$ we define $$d(x,X)=\max\{d(x,z):z\in X\}.$$ Let $x,y\in X$ be two (different) points of $X$. Is there a metric $d'$ on $X$ such that $d'(x,X)\neq d'(y,X)$?

Let $X$ be a compact metric space. Assume that $X$ has more than $2$ points (or even better, that $X$ is connected with more than 1 point). Given a metric $d$ on $X$ we define $$d(x,X)=\max\{d(x,z):z\in X\}.$$ Let $x,y\in X$ be two (different) points of $X$. Is there a metric $d'$ on $X$ (inducing the same topology) such that $d'(x,X)\neq d'(y,X)$?

Let $X$ be a compact metric space. Given a metric $d$ on $X$ we define $$d(x,X)=\max\{d(x,z):z\in X\}.$$ Let $x,y\in X$. Is there a metric $d'$ on $X$ such that $d'(x,X)\neq d'(y,X)$?

Let $X$ be a compact metric space. Assume that $X$ has more than $2$ points (or even better, that $X$ is connected with more than 1 point). Given a metric $d$ on $X$ we define $$d(x,X)=\max\{d(x,z):z\in X\}.$$ Let $x,y\in X$ be two (different) points of $X$. Is there a metric $d'$ on $X$ such that $d'(x,X)\neq d'(y,X)$?