This paper investigates the magnitude discontinuity for non-skew finite subsets of ℓN, establishing bounds relating the magnitude jump to skewness defect and geometric configuration. It provides an upper bound, a sharp lower bound, explicit formulas for magnitude, and a classification of discontinuity types.
Key findings
Magnitude discontinuity at non-skew finite subsets is bounded above by a function of collision graph structure and dimension N.
Discontinuities can be arbitrarily large for certain non-skew configurations.
Explicit formulas for magnitude of cubical thickenings in the non-skew case are derived.
A complete classification of discontinuity types based on the pattern of coordinate coincidences is provided.
Limitations & open questions
Further research is needed to extend these findings to more complex metric spaces and configurations.