Chebyshev distance

It is also known as chessboard distance, since in the game of chess the minimum number of moves needed by a king to go from one square on a chessboard to another equals the Chebyshev distance between the centers of the squares, if the squares have side length one, as represented in 2-D spatial coordinates with axes aligned to the edges of the board.

The Chebyshev distance between two vectors or points x and y, with standard coordinates

, their Chebyshev distance is 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 2r and are parallel to the coordinate axes.

In one dimension, all Lp 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 √2r, 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.

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 polytopes.

Nevertheless, it is true that in all finite-dimensional spaces the L1 and L∞ metrics are mathematically dual to each other.

The Chebyshev distance is sometimes used in warehouse logistics,[4] as it effectively measures the time an overhead crane takes to move an object (as the crane can move on the x and y axes at the same time but at the same speed along each axis).

It is also widely used in electronic Computer-Aided Manufacturing (CAM) applications, in particular, in optimization algorithms for these.

operating in the plane, are usually controlled by two motors in x and y directions, similar to the overhead cranes.

For the space of (real or complex-valued) functions, the Chebyshev distance generalizes to the uniform norm.