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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Plase say what is it that you mean by “translation invariant”. What is a translation in an arbitrary metric space? – José Carlos Santos Jan 21 '18 at 10:51
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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 – Sajad Rather Jan 21 '18 at 11:04
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Again: what is it that you mean by “translation invariant”? Are interested only on metrics defined on vector spaces? – José Carlos Santos Jan 21 '18 at 11:05
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i mean d(x+a,y+a) = d(x,y) – Sajad Rather Jan 21 '18 at 11:08
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And wha does $+a$ mean in an arbitrary metric space? – José Carlos Santos Jan 21 '18 at 11:09
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if we take X= |R^infinity ( R =real line) , – Sajad Rather Jan 21 '18 at 11:17
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I don't know what $\mathbb{R}^\infty$ is, nor what it has to do with the complex plane. – José Carlos Santos Jan 21 '18 at 11:19
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Let V be a vector space. Is any metric d which satisfies d(u, v) = d(u + w, v + w) for all u, v, w in V necessarily induced by some norm on V? – Ben G. Jan 21 '18 at 11:27
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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 $d(1+1,1+0)=7\neq1$.
José Carlos Santos
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