normed plane

A normed plane is a pair $({ℝ}^2,||\cdot ||)$, where the function$x\to ||x||$ is a norm.

If we define a distance function $d(x,y)=||x-y||$ thenthe metric space $({ℝ}^2,d)$ is called a Minkowski plane ora Minkowski geometry.

The classical examples of Minkowski and normed planes arethe $p$-norm ${||x||}_p={({|x_1|}^p+{|x_2|}^p)}^{1/p}$ where and the maximum or supremum norm${||x||}_{\mathrm{\infty }}=\max \{|x_1|,|x_2|\}$.