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.

→ The shortest distance between two points is a straight line.
Generally:

$$\begin{array}{rl}|ab|& =|a|\cdot |b|\\ |a+b|& \le |a|+|b|\text{(triangle inequality)}\\ ||a|-|b||& \le |a-b|\end{array}$$

This also translates to vector spaces with a norm:

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

$\Vert u+v{\Vert }_2=\Vert u{\Vert }_2+\Vert v{\Vert }_2⟺\exists \alpha \ge 0:u=\alpha v$, i.e. this equality holds if two vectors point in the same direction.

Absolute value as distance

On $\mathbb{R}$, $|x-y|$ is the distance between $x$ and $y$. The direct distance from $x$ to $z$ is at most the distance going via any intermediate point $y$:

$$|x-z|\le |x-y|+|y-z|$$