# normed plane

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

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

The classical examples of Minkowski and normed planes are the $p$-norm $||x||_{p}=(|x_{1}|^{p}+|x_{2}|^{p})^{1/p}$ where $1\leq p<\infty$ and the maximum or supremum norm $||x||_{\infty}=\max\{|x_{1}|,|x_{2}|\}$.

Title normed plane NormedPlane 2013-03-22 16:50:30 2013-03-22 16:50:30 Mathprof (13753) Mathprof (13753) 6 Mathprof (13753) Definition msc 46B20 Minkowski plane Minkowski geometry