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 $$\R$$ 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 $$\R^n$$.

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

Let the metric $$d_1$$ be imposed on $$\R^n$$ 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.

Graphical Example
This diagram shows the $\epsilon$-neighborhood of point $$A$$ in the $$\left({\R^2, d_1}\right)$$ metric space.


 * TaxicabMetric.png

Note that $$\epsilon = \epsilon_1 + \epsilon_2$$.

Neither the boundary lines nor the extreme points are actually part of the neighborhood.

Why the name?
Imagine a city whose streets form a grid pattern. That is, all the streets run, for example, either North-South or East-West.

It is assumed that one can travel only along the streets.

To travel between any two locations in the city, one must therefore travel a certain distance (possibly zero) North or South, and a certain distance (also possbly zero) East or West.

The driver of a taxicab constantly needs to know the distance between any two points in the city.

However, that distance is measured not directly, but along the streets of the city.

Hence the distance between any two points in a taxicab metric is measured as the sum of the difference between the corresponding coordinates of those points.