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While reading about metric spaces, the following question struck me. We know the following definition of pseudometric spaces and metric spaces:

Suppose $d: X \times X \rightarrow \mathbb{R}$ and that for all $x,y,z \in X$:

$1. d(x,y) \geq 0$

$2. d(x,x)=0$

$3. d(x,y)=d(y,x)\space\space\space\space\space$ (Symmetry)

$4. d(x,z) \leq d(x,y)+d(y,z)$ (Triangle Inequality)

Such a "distance function" $d$ is called a pseudometric on X. The pair $(X,d)$ is called a pseudometric space.

If $d$ satisfies:

$5.$ when $x \neq y,$ then $d(x,y)>0$,

then $d$ is called a metric on X and $(X,d)$ is called a metric space.

Now, $\ell_2^2$ with $d: \ell_2^2 \times \ell_2^2 \rightarrow \mathbb{R}$ violates the property of triangle inequality. Any pseudometric space $(X,d)$ would violate the non-negativity of metric spaces, since they have at least two points $x \neq y$ for which $d(x,y)=0$.

Similarly, are there any "spaces" that violate symmetry of metric spaces? If not, how do we justify mathematically?

Thank you in advance.

deditus
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  • Related: Examples of non symmetric distances. – Martin Sleziak Dec 27 '17 at 17:06
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    My own opinion is that the asymmetric case should be considered the norm, and the symmetric case exceptional. Consider for example $d(x, y) =$ work it takes to get from $x$ to $y$ in a mountainous $X$. This example was suggested by Lawvere in his metric spaces paper. For one mathematically significant example: the right adjoint $\rho_x$ to the map $[0, \infty) \to [0, \infty)$ sending $y \mapsto x + y$ is given by $z \mapsto x - z$ if $x \geq z$, else $0$. The map $(x, z) \mapsto \rho_x(z)$ is a Lawvere metric. (I do strongly recommend Lawvere's article.) – user43208 Oct 08 '19 at 23:06

4 Answers4

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There are two closely related classes of asymmetric metric spaces that come to mind, although they are not something you would encounter until, say, an upper level graduate course on low dimensional geometric topology. Namely:

  1. The Teichmuller space of a surface equipped with Thurston's asymmetric log Lipschitz metric;
  2. The outer space of a free group equipped with the asymmetric log Lipschitz metric.
Lee Mosher
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    Thank you so much :)... I tried to construct an example myself, dunno if it falls into this category or is a valid example...$d: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ such that $d(x,y)=Min[1,|x-y|]$ if $x \geq y$ and $d(x,y)=1$ if $x<y$... – deditus Mar 10 '16 at 13:40
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There is an entire area of geometry which deals with asymmetric metrics, it is called Finsler geometry. A Finsler metric on a smooth (connected) manifold $M$ is given by a choice of a function $F$ on tangent spaces of $M$ satisfying certain restrictions. Then the Finsler distance between points $p, q$ in $M$ is defined as $$ d_F(p,q)=\inf_c \int_{0}^{1} F(c'(t))dt, $$ where the infimum is taken over all paths $c: [0,1]\to M$ connecting $p$ to $q$. This Finsler distance function satisfies all the axioms of a metric except for the symmetry.

A Finsler metric is reversible if $F(-v)=F(v)$. Reversible metrics result in symmetric distance functions.

Moishe Kohan
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The Kullback Leibler divergence between two probability distributions tells you how much information samples from one distribution give you to reject the assumption that the samples come actually from the other distribution. There is no reason this should be symmetric and indeed it isn't. The other assumptions of a metric are satisfied though.

Bananach
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  • Can you please elaborate a little more? Since I haven't read about this yet, I am not able to understand your example. It would be very helpful if you add any reading source too... Thanks :) – deditus Mar 06 '16 at 09:45
  • I can't give a full exposition here. My favorite reference is Kullback's "Information Theory and Statistics". It is accessible (the first pages, which explain the KL divergence at least) with only little knowledge of probability. – Bananach Mar 06 '16 at 10:22
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    @deditus my answer is incorrect, KL divergence does not satisfy the triangle inequality. You might want to unaccept the answer – Bananach Mar 07 '16 at 06:22
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Metric spaces where distances are in $[0,\infty]$ and which drop symmetry but still satisfy the triangle inequality and and $d(x,x)\ge 0$ are called generalised metric spaces in the influential paper by Lawvere http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html

A nice example of a space where asymmetry is important is given by finite or infinite words over an alphabet, see Rutten http://www.cwi.nl/~janr/papers/files-of-papers/1996-tcs170.pdf . The idea is that $d(w,v)=0$ means that the word $w$ is a prefix of the word $v$ and that $d(w,v)=2^{-n}$ if the first letter where $w$ differs from $v$ is at position $n+1$.

Ok, this is an ultrametric space. But as the question was about asymmetry, it is still a nice example. And I think that metric space examples can be constructed along the same lines.

Alexander Kurz
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