# Definition:Taxicab Metric/Real Vector Space

## Definition

Let $\R^n$ be a real vector space.

The **taxicab metric** on $\R^n$ is defined as:

- $\ds \map {d_1} {x, y} := \sum_{i \mathop = 1}^n \size {x_i - y_i}$

where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.

## Graphical Example

This diagram shows the open $\epsilon$-ball $\map {B_\epsilon} {A; d_1}$ of point $A$ in the $\struct {\R^2, d_1}$ metric space where $d_1$ is the taxicab metric.

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

Neither the boundary lines nor the extreme points are actually part of the open $\epsilon$-ball.

## Linguistic Note

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.

## Also see

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces: Exercise $2$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.3$