Definition:Chebyshev Distance/General Definition

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Definition

Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces.

Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$.


The Chebyshev distance on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_i$ is defined as:

$\displaystyle \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, y_i} }$

where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \mathcal A$.


Real Vector Space

This metric is usually encountered in the context of the real vector space $\R^n$:


The Chebyshev distance on $\R^n$ is defined as:

$\displaystyle \map {d_\infty} {x, y}:= \max_{i \mathop = 1}^n \set {\size {x_i - y_i} }$

where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.


Also known as

The Chebyshev distance is also known as the maximum metric.

Another term is the chessboard distance, as it can be illustrated on the real number plane as the number of moves needed by a chess king to travel from one point to the other.


Some sources refer to this as the standard procedure for defining a distance function on a cartesian product of metric spaces.


Also see

  • Results about the Chebyshev distance can be found here.


Source of Name

This entry was named for Pafnuty Lvovich Chebyshev.


Sources