Exact formula for Δk′(α(k−1)) for 4 ≤ k and α ≤ k−1

Prove that for integers k ≥ 4 and α ≤ k−1, Δk′(α(k−1)) = 1/(2(α−1)), where Δk′(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 k points from the set.

Background

For general k-gons, the authors extend their method and provide upper bounds of the form (k−2)/n + o(1/n). They observe a lower bound construction for certain n obtained by placing points at the corners of an α × (k−1) grid, which guarantees Δk′(α(k−1)) ≥ 1/(2(α−1)).

They conjecture that this lower bound is tight for all α ≤ k−1, which would imply that the grid placement is optimal and, in turn, yield an improved global upper bound of approximately k/(2n) for Δk(n).