Chebyshev distance

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In mathematics, the Chebyshev distance (or Tchebychev distance) between two points in a vector space is the greatest of their differences along any coordinate dimension.[1]

Mathematically, the Chebyshev distance between two vectors or points p and q, with standard coordinates pi and qi, respectively, is

D_{\rm Chess} = \max_i(|p_i - q_i|) = \lim_{k \to \infty} \bigg( \sum_{i=1}^n \left| p_i - q_i \right|^k \bigg)^{1/k}.

The Chebyshev distance is in fact a special case of the supremum norm, and is also known as the L∞ metric.[2] It is also known as chessboard distance.[3] It is an example of an injective metric.

In two dimensions, i.e. plane geometry, if the points p and q have Cartesian coordinates (x1,y1) and (x2,y2), this becomes

D_{\rm Chess} = \max \left ( \left | x_2 - x_1 \right | , \left | y_2 - y_1 \right | \right ) .

The "circle" of radius r in the Chebyshev metric, that is, the set of points at a distance r from a center point, is a square with side length 2r parallel to the coordinate axes. The two dimensional Manhattan distance also has circles in the form of squares, with side length √2r, at an angle of π/4 to the coordinate axes, so the planar Chebyshev distance can be viewed as equivalent by rotation and scaling to the planar Manhattan distance. However this equivalence between L1 and L∞ metrics does not generalize to higher dimensions.

The Chebyshev distance is named after Pafnuty Chebyshev. In chess, the distance between squares, in terms of moves necessary for a king, is given by the Chebyshev distance, hence the second name.

The Chebychev distance is widely used in warehouse logistics.


  1. ^ James M. Abello, Panos M. Pardalos, and Mauricio G. C. Resende (editors) (2002). Handbook of Massive Data Sets. Springer. ISBN 1402004893. 
  2. ^ Cyrus. D. Cantrell (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 0521598273. 
  3. ^ David M. J. Tax, Robert Duin, and Dick De Ridder (2004). Classification, Parameter Estimation and State Estimation: An Engineering Approach Using MATLAB. John Wiley and Sons. ISBN 0470090138. 


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