Definition:Taxicab Metric

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

Let $A_{1'} \times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$.

The taxicab metric on $A_{1'} \times A_{2'}$ is defined as:


 * $\map {d_1} {x, y} := \map {d_{1'} } {x_1, y_1} + \map {d_{2'} } {x_2, y_2}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$.

Real Number Plane
This metric is often seen in the context of the real number plane $\R^2$ and general real vector space $\R^n$:

Also see

 * Taxicab Metric is Metric

Notation
The notation is awkward, because it is necessary to use a indexing subscript for the $n$ metric spaces contributing to the product, and also for the $p$th exponential that defines the metric itself.

Thus the "prime" notation on the $n$ metric spaces.