Editing Ball (mathematics) (section)

==={{mvar|p}}-norm===
In a [[Cartesian space]] {{math|'''R'''<sup>''n''</sup>}} with the [[p-norm|{{mvar|p}}-norm]] {{mvar|L<sub>p</sub>}}, that is one chooses some <math>p \geq 1</math> and defines<math display="block">\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dots + |x_n|^p \right) ^{1/p},</math>Then an open ball around the origin with radius <math>r</math> is given by the set
<math display="block"> B(r) = \left\{ x \in \R^n \,:\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dots + |x_n|^p \right) ^{1/p} < r \right\}.</math>For {{math|1=''n'' = 2}}, in a 2-dimensional plane <math>\R^2</math>, "balls" according to the {{math|''L''<sub>1</sub>}}-norm (often called the ''[[Taxicab geometry|taxicab]]'' or ''Manhattan'' metric) are bounded by squares with their ''diagonals'' parallel to the coordinate axes; those according to the {{math|''L''<sub>∞</sub>}}-norm, also called the [[Chebyshev distance|Chebyshev]] metric, have squares with their ''sides'' parallel to the coordinate axes as their boundaries. The {{math|''L''<sub>2</sub>}}-norm, known as the Euclidean metric, generates the well known disks within circles, and for other values of {{mvar|p}}, the corresponding balls are areas bounded by [[Lamé curve]]s (hypoellipses or hyperellipses).

For {{math|1=''n'' = 3}}, the {{math|''L''<sub>1</sub>}}-balls are within octahedra with axes-aligned ''body diagonals'', the {{math|''L''<sub>∞</sub>}}-balls are within cubes with axes-aligned ''edges'', and the boundaries of balls for {{mvar|L<sub>p</sub>}} with {{math|''p'' > 2}} are [[superellipsoid]]s. {{math|1=''p'' = 2}} generates the inner of usual spheres.

Often can also consider the case of <math>p = \infty</math> in which case we define <math display="block">\lVert x \rVert_\infty = \max\{\left|x_1\right|, \dots, \left|x_n\right|\}</math>