Tighten the bound on the number of 4-contact placements to O(k n)

Establish whether, for a convex k-gon P and a convex n-gon Q in the plane (under general position), the number of similar copies of P contained in Q that have four distinct vertices of P in contact with the boundary of Q can be bounded by O(k n), improving the current O(k^4 n) upper bound.

Background

The authors prove a near-linear-in-n combinatorial upper bound and an enumeration procedure for placements where four vertices of P are in contact with Q’s boundary (the 4-contact case), obtaining an O(k^4 n) bound (up to logarithmic factors) under general position. This disproves a prior conjecture expecting Ω(k n^2) such placements.

They explicitly ask whether this bound can be further tightened to O(k n), which would significantly improve the structural understanding of feasible 4-contact placements and potentially lead to faster algorithms.