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 $2.2.7$