Editing Chebyshev distance (section)

== Properties ==
[[File:Minkowski_distance_examples.svg|thumb|Comparison of Chebyshev, Euclidean and Manhattan distances for the hypotenuse of a 3-4-5 triangle on a chessboard]]
In one dimension, all L<sub>''p''</sub> metrics are equal – they are just the absolute value of the difference.

The two dimensional [[Manhattan distance]] has "circles" i.e. [[level sets]] in the form of squares, with sides of length {{sqrt|''2''}}''r'', oriented at an angle of π/4 (45°) to the coordinate axes, so the planar Chebyshev distance can be viewed as equivalent by rotation and scaling to (i.e. a [[linear transformation]] of) the planar Manhattan distance.

However, this geometric equivalence between L<sub>1</sub> and L<sub>∞</sub> metrics does not generalize to higher dimensions. A [[sphere]] formed using the Chebyshev distance as a metric is a [[cube]] with each face perpendicular to one of the coordinate axes, but a sphere formed using [[Manhattan distance]] is an [[octahedron]]: these are [[dual polyhedra]], but among cubes, only the square (and 1-dimensional line segment) are [[self-dual polyhedra|self-dual]] [[polytope]]s. Nevertheless, it is true that in all finite-dimensional spaces the L<sub>1</sub> and L<sub>∞</sub> metrics are mathematically dual to each other.

On a grid (such as a chessboard), the points at a Chebyshev distance of 1 of a point are the [[Moore neighborhood]] of that point.

The Chebyshev distance is the limiting case of the order-<math>p</math> [[Minkowski distance]], when <math>p</math> reaches [[infinity]].