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Is every metric translation invariant,if no then what are the conditions under which a metric may become translation invariant (if any) .

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  • $\begingroup$ Plase say what is it that you mean by “translation invariant”. What is a translation in an arbitrary metric space? $\endgroup$ – José Carlos Santos Jan 21 '18 at 10:51
  • $\begingroup$ since every metric induced by norm satisfy translation invariance property but i am confused about ,can we impose any condition on metrics so that they might satisfy translation invariance property $\endgroup$ – Sajad Rather Jan 21 '18 at 11:04
  • $\begingroup$ Again: what is it that you mean by “translation invariant”? Are interested only on metrics defined on vector spaces? $\endgroup$ – José Carlos Santos Jan 21 '18 at 11:05
  • $\begingroup$ i mean d(x+a,y+a) = d(x,y) $\endgroup$ – Sajad Rather Jan 21 '18 at 11:08
  • $\begingroup$ And wha does $+a$ mean in an arbitrary metric space? $\endgroup$ – José Carlos Santos Jan 21 '18 at 11:09
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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 $d(1+1,1+0)=7\neq1$.

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