# Definition:Chebyshev Distance

## Definition

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Let $A_1 \times A_2$ be the cartesian product of $A_1$ and $A_2$.

The **Chebyshev distance** on $A_1 \times A_2$ is defined as:

- $\map {d_\infty} {x, y} := \max \set {\map {d_1} {x_1, y_1}, \map {d_2} {x_2, y_2} }$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_1 \times A_2$.

### General Definition

The **Chebyshev distance** on $\ds \AA = \prod_{i \mathop = 1}^n A_i$ is defined as:

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

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

### Real Number Plane

This metric is usually encountered in the context of the real number plane $\R^2$:

The **Chebyshev distance** on $\R^2$ is defined as:

- $\map {d_\infty} {x, y}:= \max \set {\size {x_1 - y_1}, \size {x_2 - y_2} }$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.

## Graphical Example

This diagram shows the open $\epsilon$-ball $\map {B_\epsilon} {A; d_\infty}$ of point $A$ in the $\struct {\R^2, d_\infty}$ metric space where $d_\infty$ is the Chebyshev distance.

Neither the boundary lines nor the extreme points are actually part of the open $\epsilon$-ball.

## Also known as

The **Chebyshev distance** is also known as the **maximum metric** or **sup 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.

## Also see

- Results about
**the Chebyshev distance**can be found here.

## Source of Name

This entry was named for Pafnuty Lvovich Chebyshev.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.7$