It is known that a metric space is called non-Archimedean (ultra-metric) if $d(x,y)\le\max(d(x,z),d(z,y)).$ Just out of curiosity, does there exist any metric concept such that $\min(d(x,z),d(z,y))\le d(x,y)~\forall x, y, z?.$ Will imposing such condition on metric spaces lead to any contradiction?
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1What happens if you take $x=y$? – Kavi Rama Murthy Nov 13 '19 at 11:51
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I got it. Thanks – Jave Nov 13 '19 at 11:58