Editing Chebyshev distance (section)

== Definition ==
The Chebyshev distance between two vectors or points ''x'' and ''y'', with standard coordinates <math>x_i</math> and <math>y_i</math>, respectively, is

:<math>D(x,y) = \max_i(|x_i -y_i|).\ </math>
This equals the limit of the [[Lp space|L<sub>''p''</sub> metrics]]:
:<math>D(x,y)=\lim_{p \to \infty} \bigg( \sum_{i=1}^n \left| x_i - y_i \right|^p \bigg)^{1/p},</math>
hence it is also known as the L<sub>∞</sub> metric.

Mathematically, the Chebyshev distance is a [[metric (mathematics)|metric]] induced by the [[supremum norm]] or [[uniform norm]]. It is an example of an [[injective metric space|injective metric]].

In two dimensions, i.e. [[plane geometry]], if the points ''p'' and ''q'' have [[Cartesian coordinates]]
<math>(x_1,y_1)</math> and <math>(x_2,y_2)</math>, their Chebyshev distance is

:<math>D_{\rm Chebyshev} = \max \left ( \left | x_2 - x_1 \right | , \left | y_2 - y_1 \right | \right ) .</math>

Under this metric, a [[circle]] of [[radius]] ''r'', which is the set of points with Chebyshev distance ''r'' from a center point, is a square whose sides have the length 2''r'' and are parallel to the coordinate axes.

On a chessboard, where one is using a ''discrete'' Chebyshev distance, rather than a continuous one, the circle of radius ''r'' is a square of side lengths 2''r,'' measuring from the centers of squares, and thus each side contains 2''r''+1 squares; for example, the circle of radius 1 on a chess board is a 3&times;3 square.