For a homework problem, I need to find a recursive equation that relates the volume of an $n$-dimensional ball $V_n(r)$ of radius $r$ to that of an $(n-2)$-dimensional ball, expressed by $V_{n-2}(r)$. The norm for these balls is the $L^1$ norm given by $$||(x_1, \ldots, x_n)||_{\mathbb R^n,1} = |x_1| + \cdots + |x_n|$$
I would like to use multivariate change of variables to solve this problem since the previous one (basically the same except using the standard Euclidean norm) can be worked out the same way. In particular, the change of variables used for that problem is the following: $$g(r, \theta, t_3, \ldots, t_n) = (r \cos\theta, r \sin\theta, t_3,\ldots,t_n)$$
for which the derivative matrix $\mathrm Dg$ is a block matrix consisting of the standard $2 \times 2$ rotation transformation matrix at the top left, the $(n-2) \times (n-2)$ identity matrix at the bottom right, and zero everywhere else. In particular, $\det \mathrm Dg = r$. From there, one can apply this change of variables on the integral defining the volume of the unit ball (integral over 1 on the ball itself) and get the well-known integral shown here.
Now, it's possible that this same change of variables works in $L^p$ norms, but I'm not really sure. Any help would be greatly appreciated.