Sup metric on R^n

Example

Let $n \ge 1$ be an integer. For $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let $$d_\infty(x, y) = \max_{1 \le i \le n} |x_i - y_i|.$$ Then $d_\infty$ is a metric on $\mathbb{R}^n$, the sup metric.

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 $p \to \infty$.

Explanations

L1 human

The sup metric takes only the largest difference of the coordinates. On a chessboard, it is the number of moves that a king needs to go from one square to another.