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 $\stackrel{\mathrm{g}}{\mathrm{s}}(\overrightarrow{x})\cdot \overrightarrow{x}$ where $\overrightarrow{sgn}$ 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).