How to prove triangle inequality for the metric $d(x,y)=\frac{2|x-y|}{\sqrt{\left(1+|x|^2\right)}+\sqrt{\left(1+|y|^2\right)}}$

Asked Jul 24 '18 at 08:18

Active Jul 24 '18 at 08:27

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How to prove triangle inequality for the metric $$d(x,y)=\frac{2|x-y|}{\sqrt{1+|x|^2}+\sqrt{1+|y|^2}},$$ for all $x,y\in \mathbb{C}?$

edited Jul 24 '18 at 08:27

gebruiker

asked Jul 24 '18 at 08:18

Rajib Ghosh

What is the motivation for this? Hyperbolic geometry on the Poincaré disc model? – Parcly Taxel Jul 24 '18 at 08:20
This seems to be difficult. All answers to https://math.stackexchange.com/q/1104145/42969 are only partial solutions (or wrong). – Martin R Jul 24 '18 at 08:26
Are you sure that it is a metric? For example, should the radicals in the denominator he multiplied rather than added? No typo here? – MPW Jul 24 '18 at 08:32
... in that case it would be a duplicate of https://math.stackexchange.com/q/1090222/42969 or https://math.stackexchange.com/q/1159327/42969. – Martin R Jul 24 '18 at 08:49

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