Definition:Chebyshev Distance/Real Number Plane
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Definition
Let $\R^2$ be the real number plane.
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 see
- 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: Examples $2.2.3$