Definition:Chebyshev Distance/General Definition

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 $\R^n$:

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

 * Chebyshev Distance is Metric


 * Definition:Taxicab Metric
 * Definition:$p$-Product Metric