Note: An alternative proof of the vulnerability of $k$-NN classifiers in high intrinsic dimensionality regions

This document proposes an alternative proof of the result contained in article [1]. The proof is simpler to understand (I believe) and leads to a more precise statement about the asymptotical distribution of the relative amount of perturbation. Suppose that an artificial intelligent program bases its decision on the collection points neighbouring the query. Suppose that this is not the case for that query q and this collection point x . We are interested in the amount of perturbation to be applied to collection point x so that the program takes it into account.