Exact value of Δ4′(9)

Establish whether Δ4′(9) equals 1/4, where Δ4′(n) denotes the supremum, over all n-point sets placed in any unit-area rectangle of the form [0,d] × [0,d^{-1}] with 0 < d ≤ 1, of the minimum area of the convex hull of any four points from the set.

Background

In the quadrilateral case (k=4), the authors improve the trivial upper bound on Δ4(n) and analyze specific small n to tighten constants. They show via a construction (Figure 1) that Δ4′(9) ≥ 1/4 by placing points so that any four-point convex hull can be trapped within area 1/4.

They conjecture that this construction is optimal, i.e., the reverse inequality also holds, which would yield Δ4′(9) = 1/4. If true, this would immediately imply a sharper global bound Δ4(n) ≤ 2/(n−8) by choosing n′ = 9 in their pigeonhole-based reduction (Observation 2.1).