Characterization of modified cutting gaps

Characterize the properties of the modified cutting gaps and describe point sets for which the inequality τₖ,ₛ < k − s holds for at least some values of k and s.

Background

The modified cutting gap τₖ,ₛ is introduced to obtain more refined information when k-dimensional subspaces disjoint from a point set make the ordinary cutting gap less informative. The paper notes that the expected value is generally k − s, but Example 4.4 exhibits a point set for which the inequality is strict for a suitable choice of parameters. The authors explicitly identify as future work both the characterization of the modified cutting-gap properties and the classification or description of sets exhibiting such strict inequalities.