triangle inequality

Triangle Inequality

For any triangle, the sum of the lenghts of any two sides must be greater than or equal to the length of the remainingg side:

$$c\le a+b$$

With equality only in the degenerate case of a triangle with zero area.

Triangle inequality in euclidian geometry (amongst others)

The lengths of sides are replaced by vector norms of vectors $\mathbf{u},\mathbf{v}\in {\mathbb{R}}^1$

$$\Vert \mathbf{u}+\mathbf{v}\Vert \le \Vert \mathbf{u}\Vert +\Vert \mathbf{v}\Vert$$

This inequality expressses the fact that the shortest distance between two points is a straight line.