L1 norm

Definition

Suppose $n$ is a positive integer. The $L^{1}$-norm, denoted $\|\cdot \|_{1}$, is a function from $\mathbb {R} ^{n}$ to $\mathbb {R}$ defined as:

$\|x\|_{1}=\sum _{i=1}^{n}|x_{i}|$

Definition in terms of the signum vector function

The $L^{1}$-norm $\|x\|_{1}$ can be defined as the dot product ${\overline {\operatorname {sgn} }}({\vec {x}})\cdot {\vec {x}}$ where ${\overline {\operatorname {sgn} }}$ denotes the signum vector function.

Definition as a piecewise linear function

The $L^{1}$-norm can be defined as a piecewise linear function. The number of pieces of the domain involved is $2^{n}$ interior regions, and an additional $3^{n}-2^{n}$ boundary regions and boundary intersections (these can be included in the piece definitions for any of the bordering interior regions).