triangle inequality

These, in turn follow with a little effort from the triangle inequality.

Taking the square root of the final result gives the triangle inequality.

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.

This follows from the triangle inequality and homogeneity of the norm.

It is easy to construct an example which disproves the property of triangle inequality.

Furthermore, we have the following generalization of the triangle inequality:

Some authors work with a weaker form of the triangle inequality, such as:

The second requirement states that none of these distances can be reduced without violating the triangle inequality.

Put otherwise, the edge weights satisfy the triangle inequality.

The triangle inequality expresses the fact that detours aren't shortcuts.

The triangle inequality is a defining property of norms and measures of distance.

The triangle inequality for general values of p is called Minkowski's inequality.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence.

Every norm is a convex function, by the triangle inequality and positive homogeneity.

It's necessary and sufficient for a heuristic to obey the triangle inequality in order to be consistent.

Furthermore in order to represent the sides of a triangle they must satisfy the triangle inequality.

For example, inputs to the general Travelling salesman problem need not obey the triangle inequality, unlike real road networks.

Also it is not hemimetric, i.e., the triangle inequality does not hold, except in special cases.

Also, strictly speaking F is not a genetic distance, as it does not satisfy the triangle inequality.

A field or normed space satisfying the ultrametric triangle inequality is called non-Archimedean.

It is never larger than the Ahlswede-Winter value (by the norm triangle inequality), but can be much smaller.

(But note that for the diameter of the symmetric difference the triangle inequality does not hold.)

The intuition behind such seemingly strange effects is that, due to the strong triangle inequality, distances in ultrametrics do not add up.

The proof for the reverse triangle uses the regular triangle inequality, and :