# 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 $\R^n$:

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

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.2$: Metric Spaces: Theorem $2.3$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.2$: Examples: Example $2.2.7$