TITLE: Spaces with a quasi triangle inequality
QUESTION [8 upvotes]: How do you call a space with a function which is symmetric, non negative, positive definite and which satisfies a quasi-triangle inequality:
$d(x,z) \leq C( d(x,y)+d(y,z) )$
for all $x,y,z$ and some $C > 1$?
That is, it satisfies all the axioms of a metric space except for the triangle inequality, which is replaced by the one above.
Can anyone provide any reference on these spaces?
Thanks.

REPLY [5 votes]: Here is a negative answer for your additional remark concerning Banach's fixed point theorem: Consider $d(x,y)=(\int_0^1|x-y|^p)^{1/p},$ $0<p<1$ which satisfies the quasi-triangle inequality. Look at the set of all (measurable real) functions on $[0,1]$ boundaed between 0 and 2 and of integral 1. Look at the Baker transformation on this set: first map $x$ to $y(t)=2x(2t)$, $0\le t\le 1/2$ then trancate at hight 2 and shift what remains ($(y-1)^{+}$) by $1/2$ to the right and $2$ down. I think I checked that it is a contraction (with constant $2^{p-1}$). This map is known not to have a fixed point, see the following paper of Dale Alspach: 
http://www.claremontmckenna.edu/math/moneill/Math%20138/papers138/Alspach.pdf