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 $ (X,\rho_x)$ and $ (Y,\rho_y)$ be two metric spaces. A metric space $ (Z,\rho_z)$ can be constructed for the Cartesian product $ Z = X \times Y$ by defining the metric $ \rho_z$ as

$\displaystyle \rho_z(z,z') = \rho_z(x,y,x',y') = c_1 \rho_x(x,x') + c_2 \rho_y(y,y'),$ (5.4)

in which $ c_1 > 0$ and $ c_2 > 0$ are any positive real constants, and $ x,x' \in X$ and $ y,y' \in Y$. Each $ z \in Z$ is represented as $ z =

Other combinations lead to a metric for $ Z$; for example,

$\displaystyle \rho_z(z,z') = \Big(c_1 \big[\rho_x(x,x')\big]^p + c_2 \big[\rho_y(y,y')\big]^p \Big)^{1/p},$ (5.5)

is a metric for any positive integer $ p$. 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