Consider a random Poisson process on arbitrarily large volume in $R^d$ enclosed by an $R^{(d-1)}$ dimensional sphere. The process terminates when a density of points $\rho$ is achieved (letting $N$ equal the number of points, we have $\rho = \frac{N}{V_{d}(R)}$). Is there a nice closed-form expression for the the minimum distance, $l_{NN}$ in $d$ dimensions that is larger than the closest nearest-neighbor distance for some fraction, $f$, of the points? In lieu of this, what if we concern ourselves with the median closest nearest-neighbor distance? I suppose it would be a fair assumption that the closest nearest neighbor distance should decay exponentially as a function of $\rho$.
2
-
$\begingroup$ I think there are some issues with your problem statement. You cannot sample points with uniform distribution in $\mathbb R^d$. Also, what do you mean by "achieving a density"? $\endgroup$– Joris BierkensCommented Nov 5, 2013 at 13:07
-
$\begingroup$ @JorisBierkens Let me attempt to fix the problems. $\endgroup$– SMellonCommented Nov 5, 2013 at 13:09
Add a comment
|