$L_p$ metrics

The most important family of metrics over ${\mathbb{R}}^n$ is given for any $p \geq 1$ as

$$\rho(x,x^\prime) = \bigg( \sum_{i=1}^n \vert x_i - x_i^\prime\vert^p \bigg)^{1/p} .$$ (5.1)

For each value of $p$, (5.1) is called an $L_p$ metric (pronounced ``el pee''). The three most common cases are

1. $L_2$: The Euclidean metric, which is the familiar Euclidean distance in ${\mathbb{R}}^n$.
2. $L_1$: The Manhattan metric, which is often nicknamed this way because in ${\mathbb{R}}^2$ it corresponds to the length of a path that is obtained by moving along an axis-aligned grid. For example, the distance from $(0,0)$ to $(2,5)$ is $7$ by traveling ``east two blocks'' and then ``north five blocks''.
3. $L_\infty$: The $L_\infty$ metric must actually be defined by taking the limit of (5.1) as $p$ tends to infinity. The result is

   $$L_\infty(x,x^\prime) = \max_{1 \leq i \leq n} \{ \vert x_i - x^\prime_i\vert \},$$ (5.2)

   
   
   which seems correct because the larger the value of $p$, the more the largest term of the sum in (5.1) dominates.

An $L_p$ metric can be derived from a norm on a vector space. An $L_p$ norm over ${\mathbb{R}}^n$ is defined as

$$\Vert x\Vert _p = \bigg( \sum_{i=1}^n \vert x_i\vert^p \bigg)^{1/p} .$$ (5.3)

The case of $p = 2$ is the familiar definition of the magnitude of a vector, which is called the Euclidean norm. For example, assume the vector space is ${\mathbb{R}}^n$, and let $\Vert\cdot\Vert$ be the standard Euclidean norm. The $L_2$ metric is $\rho(x,y) = \Vert x-y\Vert$. Any $L_p$ metric can be written in terms of a vector subtraction, which is notationally convenient.