= Solution
Let $N$ be the homogeneous <Poisson point process> in $\mathbb R^3$ with intensity measure $\lambda\,dx$, and map each point $x$ to its radius $g(x)=|x|$. For an interval $(a,b]\subseteq\mathbb R_+$, the <pushforward measure> of the intensity is
$$
\lambda\,\operatorname{vol}\{x:a<|x|\leq b\}
=\frac{4\pi\lambda}{3}(b^3-a^3).
$$
The <Mapping theorem for Poisson point processes> therefore shows that the radii form a Non-homogeneous Poisson point process with intensity function
$$
\rho(r)=\frac{d}{dr}\left(\frac{4\pi\lambda r^3}{3}\right)=4\pi\lambda r^2,
\qquad r>0.
$$
If $R$ is the distance to the closest star, the event $\{R>r\}$ says that the ball of radius $r$ contains no points. The zero-count probability of a <Poisson distribution> gives
$$
\mathbb P(R>r)=\exp\left(-\frac{4\pi\lambda r^3}{3}\right).
$$
Differentiating its <cumulative distribution function> yields the <probability density function>
$$
\boxed{f_R(r)=4\pi\lambda r^2
\exp\left(-\frac{4\pi\lambda r^3}{3}\right),
\qquad r\geq0.}
$$
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