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.


ChebyshevDistance.png


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