Asymmetric metric

A metric on a space X is a function $d$ from $X \times X$ to R satisfying the following properties

1. $d(x,y)\geq 0$
2. $d(x,y)=0$ if and only if $x=y$
3. $d(x,y)\leq d(x,z) +d(z,y)$ for all $x,y \in X$
4. $d(x,y)=d(y,x)$ for all $x,y\in X$

The standard example of this is probably the usual Euclidean metric on Euclidean space. Now it is easy to find “real life” examples in which the fourth axiom doesn’t hold, but the other three do. For example, pick any city that has at least one one-way street. Then the distance from point A to point B along that one way street will be different from B to A, unless perhaps the one way street is a circle and A and B are on opposite points on the circle. For any other pair of points, the distances are not equal.

A discrete example is a strongly connected digraph (the graph must be strongly connected, for if not, then there is a pair of vertices a and b such that there is no path from a to b, and thus the distance between the two is infinite, which contradicts that fact that the distance must be a real number).

A continuous example is a circle in which it is only possible to travel clockwise.

A function which satisfies all but condition 4 in the definition of metric is called a quasimetric (The word quasimetric may have different meanings, depending on the context). Elsewhere it is simply referred to as an asymmetric metric, or asymmetric distance, (these meaning of these terms are also inconsistent). Whatever you call it, there seems to be little of substance on the matter. The term quasimetric seems to yield the best results on Google.

Is there a candidate for a “standard” example of a (non-trivial) quasimetric space? Is there a standard quasimetric on Euclidean space?

Is there a relevant source that covers the non-trivialities of quasimetrics?

Are there any other real-life examples of quasi-metrics besides cities with one way streets?

Tags:

One Response to “Asymmetric metric”

1. quasimetric Says:

I’m compiling a list of real-life quasimetrics at the above link. Here’s what I’ve got so far: