p-metric on R^n

Example

Let $n \ge 1$ be an integer and let $p \ge 1$ be a real number. For $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let $$d_p(x, y) = \left( \sum_{i=1}^{n} |x_i - y_i|^p \right)^{1/p}.$$ Then $d_p$ is a metric on $\mathbb{R}^n$.

Remarks

The triangle inequality for $d_p$ is the Minkowski inequality. For $p = 1$ it is the taxicab metric, and for $p = 2$ it is the Euclidean metric. For $0 < p < 1$, the same formula does not give a metric when $n \ge 2$: the triangle inequality fails. The metric $d_p$ comes from the p-norm: $d_p(x, y) = \|x - y\|_p$.

Explanations

L1 human

The p-metrics form a family of ways to combine the differences of the coordinates into one distance. A larger $p$ gives more weight to the largest difference, and for very large $p$ the distance is close to the sup metric.