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For $r\ge 0$, Lebesgue measure of the ball is $v_d{r}^d$, so the Poisson random measure gives $N_r\sim \mathrm{Poisson}(v_d{r}^d)$ andBy monotonicity of the balls, the set of radii with no points is an interval starting at zero. Thus $\{R\ge r\}=\{N_r=0\}$: if all smaller balls are empty then their union, the open ball of radius $r$, is empty; if that ball is empty then its radius belongs to the defining set. We obtainThe continuous tail gives $\mathbb{P}(R=0)=0$ and $\mathbb{P}(R=\infty )=0$. Differentiating the cumulative distribution function gives the probability density functionThe substitution $u=v_d{r}^d$ proves that this density integrates to one; equivalently $v_d{R}^d$ has the unit-rate exponential distribution. This is the first-neighbour case of the Kth-nearest-neighbour distance in a homogeneous Poisson point process.

Here $\Lambda$ is planar Lebesgue measure. For a Borel set $B\subseteq [0,\infty )$, polar coordinates giveThe pushforward is therefore locally finite and atomless. By the mapping theorem for Poisson point processes, $f(\Pi )$ is a homogeneous Poisson process on $[0,\infty )$ with constant intensity $\pi$. This is the two-dimensional radial volume transform of a homogeneous Poisson point process.

Put $S_k=R_k^2$. The variable $S_k$ is the $k$th arrival time of that rate-$\pi$ Poisson process, hence has the gamma distribution with shape $k$ and rate $\pi$:Since $s=r^2$ and $ds/dr=2r$, the change-of-variables formula for a probability density gives the Kth-nearest-neighbour distance in a homogeneous Poisson point process

Put $T_k=R_k^3$. Part (i) identifies $T_k$ as the $k$th arrival time of a ratePoisson process. Hence $T_k$ has the gamma densityThe change of variables $t=r^3$, with $dt=3r^{2}dr$, gives the kth-nearest-neighbour distance in a homogeneous Poisson point process: