I know that if $(X,d_1), (X,d_2) $ are two metric spaces on $X$ then difference $d_1-d_2$ may not be a metric on $X$ as difference may give negative value also . Is there any condition that make $d_1-d_2$ a metric on $X$? Thanks in advance.
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@Paul thank ... but I want proof ... – neelkanth Jan 28 '20 at 14:15
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Actually, my statement was wrong. Please see below. – Paul Jan 28 '20 at 14:22
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For a metric space $X$ and metric $d$, $2d$ is also a metric on $X$. Let $d_1=2d$ and $d_2=d$. Then $d_1-d_2=d$ which is a metric.
Paul
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This can of course be generalized to $d_2 := (1+\varepsilon) d_1$ for all $\varepsilon > 0$. – ViktorStein Feb 19 '20 at 10:01