The $\ell_{p}$-norms are a [[02 Areas/Math/Class|class]] of [[Vector Norm|Vector Norms]] defined as the following: $\huge \lvert \lvert \vec x \rvert \rvert _{p} = \pa{ |x_{1}|^{p} + |x_{2}|^{p} + \cdots + |x_{n}|^{p} }^{\frac{1}{p}} $ For $p=2$, this is simply [[Euclidean Distance]] / the normal notion of [[Vector Magnitude]]. For $p=1$, this is equivalent to [[Taxicab Distance]]. ## Kings Distance Kings Distance, or $\ell_{\infty}$ is a special case of an [[LP Norm]] where we instead view the [[Limits|Limit]] of $p\to\infty$. This is also known as 'L-[[Infinity]] Norm'. $\huge \begin{align} ||\vec x||_{\infty} &= \pa{\lim_{ p \to \infty } \sum^{n}_{i} |x_{i}|^{p}}^{\frac{1}{p}} \end{align} $ This can be simplified into the following: $ \huge \begin{align} \ell_{\infty} : ||\vec x||_{\infty} = \max_{i} |x_{i}| \end{align} $ In other words, the [[Vector Norm|Norm]] of $\vec x$ will be the absolute value of its largest (ignoring sign) element $x_i$ .