I'm trying to understand the following assertion which is made in this paper on p. 3:
In $d$-dimensional space the point density, defined as the number of points per unit volume, is inversely proportional to the $d$-th power of the inter-point distance.
The paper is about generating a point cloud such that the point density is $p$ (and some spatial characteristics of the cloud are satisfied).
I'm only interested in $d=2$ and the domain $\Omega=[0,1)^2$. Let $p$ be a probability density function on $[0,1)^2$. What I could imagine what is being meant is the following: Let $k\in\mathbb N$ and $X_1,\ldots,X_k$ be $[0,1)^2$-valued (independent?) random variables distributed according to $p$. Define the random measure $$\zeta(B):=\left|\left\{i\in\left\{1,\ldots,k\right\}:X_i\in B\right\}\right|\;\;\;\text{for }B\in\mathcal B([0,1)^2).$$ Then $\zeta(B)$ is the number of (random) points $X_i$ in $B\in\mathcal B([0,1)^2)$.
Now what is the "point density" $\rho$? Is it a density (with respect to which measure) of $\zeta(B)$?
EDIT: Maybe I should ask in general: If samples are distributed according to $p$, what is the "point density" of the corresponding sample points?