3.5b p-Norm

p-Norm

For a vector $\vec{x}\in {\mathbb{R}}^n$, the ${\ell }^p$ norm is expressed as
$||\vec{x}|{|}_p={\left({\sum }_{i=1}^n|x_i{|}^p\right)}^{\frac{1}{p}}.$
Special cases of the $p$-norm include $p=1$ (the Manhattan norm), $p=2$ (the Euclidean norm), and $p=\infty$ (the Chebyshev norm), such that $||\vec{x}|{|}_{\infty }=\max (\vec{x})$ is the largest element of $\vec{x}$.

Then, the ${\ell }^p$ unit norm ball is defined as the set of all vectors with ${\ell }^p$ less than or equal to one,
${\mathcal{B}}^p=\{\vec{x}\in {\mathbb{R}}^n:\Vert \vec{x}{\Vert }_p\le 1\},$
and are useful for describing the geometry of the space under the ${\ell }^p$  norm. Given are illustrations for the respective spaces of the ${\ell }^1$, ${\ell }^2$ and ${\ell }^{\infty }$ in ${\mathbb{R}}^2$:

Properties

Examples