structural:scholar_edition, quality:correction, linguistic:voice_change, stylistic:tone, structural:reformatting
p -norm

In a Cartesian space R with the p -norm L, that is one chooses some p ≥ 1 {\displaystyle p\geq 1} and defines ‖ x ‖ p = (| x 1 | p + | x 2 | p + ⋯ + | x n | p) 1 / p, {\displaystyle \left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p},} Then an open ball around the origin with radius r {\displaystyle r} is given by the set B (r) = { x ∈ R n: ‖ x ‖ p = (| x 1 | p + | x 2 | p + ⋯ + | x n | p) 1 / p < r }. {\displaystyle B(r)=\left\{x\in \mathbb {R} ^{n}\,:\left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p}<r\right\}.} For n = 2, in a 2-dimensional plane R 2 {\displaystyle \mathbb {R} ^{2}}, "balls" according to the L -norm (often called the taxicab or Manhattan metric) are bounded by squares with their diagonals parallel to the coordinate axes; those according to the L -norm, also called the Chebyshev metric, have squares with their sides parallel to the coordinate axes as their boundaries. The L -norm, known as the Euclidean metric, generates the well known disks within circles, and for other values of p, the corresponding balls are areas bounded by Lamé curves (hypoellipses or hyperellipses).

For n = 3, the L - balls are within octahedra with axes-aligned body diagonals, the L -balls are within cubes with axes-aligned edges, and the boundaries of balls for L with p > 2 are superellipsoids. p = 2 generates the inner of usual spheres.

Often can also consider the case of p = ∞ {\displaystyle p=\infty } in which case we define ‖ x ‖ ∞ = max { | x 1 |, …, | x n | } {\displaystyle \lVert x\rVert _{\infty }=\max\{\left|x_{1}\right|,\dots,\left|x_{n}\right|\}}