volume of unit ball in high dimension with maximum norm If $z=(x,y)$, $x\in X, y\in Y$, space $X, Y$ could have any distance $d_x, d_y$, and space $z\in Z$ has maximum distance: $$d(z,z')=\max\{d_x(x,x'),d_y(y,y')\}.$$
If the volume of unit ball in $X, Y$ are $c_x, c_y$, the volume of unit ball in $Z$ is $$c_z=c_xc_y.$$
Why this is true? I am lost when different distances involves together. Could any body explain to me the general definition for "volume"(for a unit ball) in product space, like $Z=(X, Y)$?
Thank you very much!

The reason is that the unit ball in $Z$ is the Cartesian product of the unit balls of $X$ and $Y$. Indeed, if $z=(x,y)$, then 
$$
d(z,0)\le 1 \iff d(x,0)\le 1 \text{ and } d(y,0)\le 1 
$$
And the volume (I assume Lebesgue measure) of the product of two sets is the product of their volumes; this comes directly from the construction of the measure as the product of one-dimensional Lebesgue measures. 
