Application of classical geometrical-configuration theory

Determine whether classical geometrical-configuration theory can resolve the nonexistence of optimal constrained long-legged T arrangements for n > 3 and establish whether that theory has further applications to the region-maximization problems studied in the paper.

Background

The paper observes that the crossing graphs associated with many optimal arrangements satisfy the incidence conditions of geometrical configurations: crossing points are incident with arms, and each arm contains a prescribed number of crossing points.

For constrained long-legged T shapes, an arrangement attaining the pairwise four-intersection bound for n > 3 would produce a particular geometrical configuration. The authors ask whether the classical theory of such configurations can rule out these arrangements or otherwise provide tools for the broader family of shape-dissection problems.