#### Cartesian products of metric spaces

Metrics extend nicely across Cartesian products, which is very convenient because C-spaces are often constructed from Cartesian products, especially in the case of multiple bodies. Let and be two metric spaces. A metric space can be constructed for the Cartesian product by defining the metric as

 (5.4)

in which and are any positive real constants, and and . Each is represented as .

Other combinations lead to a metric for ; for example,

 (5.5)

is a metric for any positive integer . Once again, two positive constants must be chosen. It is important to understand that many choices are possible, and there may not necessarily be a correct'' one.

Steven M LaValle 2012-04-20