Sup metric on R^n

Remarks

It is also called the Chebyshev distance or the maximum metric. For all $x, y$ in $\mathbb{R}^n$, $d_\infty(x, y) \le d_2(x, y) \le d_1(x, y) \le n \, d_\infty(x, y)$, where $d_2$ is the Euclidean metric and $d_1$ is the taxicab metric. It is the limit of the p-metric $d_p$ as $p \to \infty$.