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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?


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More precisely, I'm trying to use Banach's fixed point theorem on this spaces, but I can only use it if the contraction constant is small enough. Is it possible to use Banach's fixed point theorem independently of how big is the contraction constant (as long as it's < 1 of course)? – John H Feb 21 '11 at 0:27
up vote 10 down vote accepted

Your construction is a special case of semimetric spaces with relaxed triangle inequality: This type of metric is sometimes also called non-Archimedian metric. There is a classical paper of W.A.Wilson "On semi-metric spaces", Amer. J. Math. 53 (1931) 361–373, on the subject. Also, I have seen this type of construction mostly used in fixed-point theory, so this would be an additional keyword to look for.

EDIT: To answer your second question about whether Banach fixed-point theorem would be applicable to semimetric spaces: In general one needs $(X,d)$ to be bounded, otherwise there are counter-examples. Consider $X=\mathbb{N}$, $d(n,m):=\frac{|n-m|}{2^{\min(n,m)}}$ and $f(n):=n+1$. Then $(X,d)$ is $d$-Cauchy complete semimetric space (!), but $f$ has no fixed points, even though it is a contraction w.r.t. $d$ with contraction constant $1/2$. This example is taken from the paper "Nonlinear Contractions on Semimetric Spaces" by J. Jachymski, J. Matkowski, T. Swiatkowski, Journal of Applied Analysis Vol. 1, No. 2 (1995), pp. 125–134, where you can also find the proof of the Banach Fixed-Point Theorem for bounded semimetric spaces and some more related results.

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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:

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This is called "C-relaxed triangular inequality". See, for example, this paper by Fagin and Stockmeyer.

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Can the Banach fixed point theorem can be used in this spaces? – John H Feb 21 '11 at 0:05
My guess is "yes" if the contraction constant is small enough comparing to $C$. But I am not sure because I never thought about it. – Mark Sapir Feb 21 '11 at 0:10

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