Taxicab metric

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 $$d_1(x, y) = \sum_{i=1}^{n} |x_i - y_i|.$$ Then $d_1$ is a metric on $\mathbb{R}^n$, the taxicab metric.

Remarks

The triangle inequality for $d_1$ follows from the triangle inequality for real numbers in each coordinate. In $\mathbb{R}^2$, the distance from $(0, 0)$ to $(3, 4)$ is $7$, but the Euclidean distance is $5$. It is the case $p = 1$ of the p-metric.

Explanations

L1 human

A taxi in a city with a grid of streets cannot drive through the blocks. It must go along the streets, so it travels the sum of the east-west and the north-south distances.