Is every metric translation invariant,if no then what are the conditions under which a metric may become translation invariant (if any) .
closed as off-topic by Namaste, Shaun, Somos, Adrian Keister, Xander Henderson Aug 19 '18 at 0:30
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Even if we are working on a vector space, the answer is negative. In $\mathbb R$, you can define the distance $d(x,y)=\bigl|x^3-y^3\bigr|$, which is not translation invariant: $d(1,0)=1$ and $d(1+1,1+0)=7\neq1$.