Definition:Chebyshev Distance/Real Vector Space

Definition

Let $\R^n$ be an $n$-dimensional real vector space.

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

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

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

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

Also see

• Chebyshev Distance on Real Vector Space is Metric

• Results about the Chebyshev distance can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.3$