= Solution
Write $r=(x^2+y^2+z^2)^{1/2}$, so $f(x,y,z)=r^3$. For $t\geq0$,
$$
f^{-1}([0,t])
=\{(x,y,z):r\leq t^{1/3}\},
$$
whose volume is
$$
\frac{4\pi}{3}\bigl(t^{1/3}\bigr)^3
=\frac{4\pi}{3}t.
$$
Thus the pushforward intensity of $[0,t]$ is
$$
\lambda\,\operatorname{vol}(f^{-1}([0,t]))
=\frac{4\pi\lambda}{3}t.
$$
The pushforward is locally finite and diffuse, so the <radial volume transform of a homogeneous Poisson point process> and the mapping theorem show that
$$
\boxed{f(\Pi)\text{ is a homogeneous Poisson process of rate }
\rho=\frac{4\pi\lambda}{3}}.
$$
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