Definition:Taxicab Metric

Metric Product Space
Let $$M_{1'} = \left\{{A_{1'}, d_{1'}}\right\}, M_{2'} = \left\{{A_{2'}, d_{2'}}\right\}, \ldots, M_{n'} = \left\{{A_{n'}, d_{n'}}\right\}$$ be a finite number of metric spaces.

Let $$\mathcal{A}$$ be the Cartesian product $$\prod_{i=1}^n \left\{{A_{i'}, d_{i'}}\right\}$$.

Let $$x = \left({x_1, x_2, \ldots, x_n}\right) \in \mathcal{A}$$ and $$y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal{A}$$.

Let the metric $$d_1$$ be imposed on $$\mathcal{A}$$ such that $$d_1 \left({x, y}\right) = \sum_{i=1}^n d_{i'} \left({x_{i'}, y_{i'}}\right)$$.

Then the product space $$\mathcal{M} = \left\{{\prod_{i=1}^n \left\{{A_{i'}, d_{i'}}\right\}, d_1}\right\}$$ is a metric space.

The metric $$d_1$$ is called the taxicab metric.

Real Vector Space
Let each of $$\left\{{A_{i'}, d_{i'}}\right\}$$ be the real number line $$\reals$$ under the usual metric.

Thus the Cartesian product $$\prod_{i=1}^n \left\{{A_{i'}, d_{i'}}\right\}$$ is the $n$-dimensional real vector space $$\reals^n$$.

Let $$x = \left({x_1, x_2, \ldots, x_n}\right) \in \reals^n$$ and $$y = \left({y_1, y_2, \ldots, y_n}\right) \in \reals^n$$.

Let the metric $$d_1$$ be imposed on $$\reals^n$$ such that $$d_1 \left({x, y}\right) = \sum_{i=1}^n \left|{x_i - y_i}\right|$$.

The metric $$d_1$$ is called the taxicab metric.