Explain the absence of local maxima in KNN and RDT core–shell analyses

Explain why fixed-number nearest-neighbor (KNN) and radical Delaunay triangulation (RDT) neighborhood definitions do not exhibit downturns at their smallest neighborhood sizes, and therefore do not show local maxima analogous to those observed for the pairwise cutoff distance (PCD) method in the core–shell analysis of D²_min.

Background

The paper evaluates how the neighborhood used to calculate the local nonaffine-motion measure D²_min affects the distinction between a central core and an adjoining shell. In the PCD analysis, the contrast between core and shell, measured through both the residual ratio Q and the difference between fitted affine matrices, exhibits clear maxima near C = 1.5–1.8, corresponding to approximately 6–8 neighbors.

At very small PCD cutoffs, the neighborhood contains too few particles, causing the core and shell affine fits to become similar and producing a downturn in the contrast measures. The authors observe that KNN and RDT do not show comparable downturns at their smallest neighborhood sizes, leaving unexplained why these methods lack corresponding local maxima.