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

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

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

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