triangle inequality

The size of a sum is at most the sum of the sizes.

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 $y$:

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