Affine-family crossing-equality conjecture

Prove that every shape belonging to the paper’s affine family can be arranged in any number of copies so that equality holds in the crossing upper bound V_x ≤ nσ(S) + binom(n,2)κ(S), thereby maximizing the total number of crossings and regions.

Background

For a shape S, σ(S) is the maximum number of self-crossings in one copy and κ(S) is the maximum number of intersections between two copies. Consequently, n copies have at most nσ(S) + binom(n,2)κ(S) crossings. Since the number of regions is maximized precisely by maximizing crossings for the straight-segment shapes under consideration, attaining this bound would yield an optimal arrangement.

The authors report that equality is achieved in most examples but identify constrained shapes and certain curved or mixed shapes as difficult. They formulate a broad conjecture for the affine family, which includes lines, hatpins, k-armed Vs, k-chains, several long-legged letters, and convex polygons.