On the performance of edited nearest neighbor rules in high dimensions

In a Poisson process in n-dimensional Euclidian space the expected distance to the kth nearest neighbor (NN) is
$E (R_{k}) = \sqrt{n \over 2 \pi e} + O \left({\rm In} \, n\over \sqrt{n} \right), \quad n \rightarrow \infty,$
and the variance is
$\sigma^{2} (R_{k}) = {1\over n} {1\over 2 \pi e} \left(\Gamma^{\prime \prime} (k)-{\Gamma'(k) \over \Gamma(k)} \right) + O \left({{\rm In} \, n \over n^{2}} \right), \quad n \rightarrow \infty.$
This implies that, for sufficiently large n, the k nearest neighbors of any point in the process lie in a thin hyperspherical shell, and that neighbors have disjoint sets of it-nearest neighbors. This makes possible the evaluation of performance for k-NN classification with edited data sets.