The last condition, 4, is called the triangle inequality.
If this is the case, there is also such an M for each other a, by the triangle inequality.
The second requirement states that none of these distances can be reduced without violating the triangle inequality.
The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence.
Furthermore, we have the following generalization of the triangle inequality:
It is easy to construct an example which disproves the property of triangle inequality.
Some authors work with a weaker form of the triangle inequality, such as:
Put otherwise, the edge weights satisfy the triangle inequality.
These, in turn follow with a little effort from the triangle inequality.
This follows from the triangle inequality and homogeneity of the norm.
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