Definition:Taxicab Metric

Definition
Let $M_{1'} = \left({A_{1'}, d_{1'}}\right)$ and $M_{2'} = \left({A_{2'}, d_{2'}}\right)$ 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:


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

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

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

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.