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 "High intrinsic dimensionality facilitates adversarial attack: Theoretical evidence", Amsaleg et a.. 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.