metric with relaxed inequality

Stream: general

[Jireh Loreaux]

This is not for current naming of a structure in mathlib, but I'm wondering what people in this community call a metric (or norm) structure which satisfies a "relaxed" triangle inequality, i.e., $d(x,z) ≤ K(d(x,y) + d(y,z))$ for some $K ≥ 1$ (or what is almost equivalent up to a factor of 2 on $K$ : $d(x,z) ≤ K(\max \{d(x,y), d(y,z)\})$).

[Jireh Loreaux]

/poll relaxed metric structure is called
quasimetric

[Heather Macbeth]

cc @Filippo A. E. Nuccio

[Damiano Testa]

In some contexts a quasi-metric space is one lacking symmetry, rather than allowing a distorsion factor in the triangle inequality.

[Jireh Loreaux]

Yes, this was the reference Yaël used also. I confused him by mentioning quasi-metrics in reference to the relaxed triangle inequality. I guess my only question is: what is the terminology we think is standard? (This is only going into a docstring comment about pi_Lp, so it's not like it phenomenally important.)

[Damiano Testa]

Honestly, I'm more used to hearing about quasi isometries, where the distortion factor makes more sense than not imposing symmetry. Maybe you could write exactly what you wrote above "a metric structure with a relaxed triangle inequality". :shrug:

[Filippo A. E. Nuccio]

I like the book An F-space sampler by Kalton, Peck, Roberts, where they study precisely this kind of objects. They call a norm that satisfies a relaxed triangle inequality (but still scales as $\Vert \alpha x\Vert=\vert\alpha\vert \Vert x\Vert$) a quasi-norm, and a complete locally bounded vector space (every topology induced by a quasi-norm is locally bounded) a quasi-Banach space. So I think that the name quasi-metric would be very appropriate (a part from the hyphen).