Definition:Taxicab Metric/General Definition

Definition

Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ be metric spaces.

Let $\ds \AA = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.

The taxicab metric on $\ds \AA = \prod_{i \mathop = 1}^n A_{i'}$ is defined as:

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

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

Real Vector Space

This metric is often seen in the context of the real vector space $\R^n$:

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

Also see

• Taxicab Metric is Metric

• Results about the taxicab metric can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.7$