Strict subadditivity of the parametric improvement f_max(k)

Prove the following strict subadditivity property for the parametric lower-bound function used in the paper. For ρ ∈ [1, 2/√3], define f(ρ, k) = (area(H) − k · area(R_2^ρ ∩ H)) / (area(R_0^ρ ∩ H) + area(R_2^ρ ∩ H)) + k, where H is the Voronoi region of the origin in the hexagonal lattice A_2 and R_j^ρ is the set of points covered by exactly j disks in the translated family of radius-ρ disks centered at A_2. Let f_max(k) = max_{ρ ∈ [1, 2/√3]} f(ρ, k) and g_max(k) = f_max(k) − f_max(0). Establish that for all nonnegative integers k1 and k2, g_max(k1 + k2) < g_max(k1) + g_max(k2). Equivalently, show that f_max(k1 + k2) < f_max(k1) + k2 for all nonnegative integers k1 and k2.

Background

The paper introduces a parametric variant of Inaba’s method by considering a family of overlapping unit-disk coverings 𝒜₂^ρ centered on the hexagonal lattice A_2, with parameter ρ ∈ [1, 2/√3]. For a point set X that includes k generalized boundary points, the authors derive a lower bound on the maximum size of X that can always be exactly covered, expressed via a function f(ρ, k). Maximizing f(ρ, k) over ρ yields f_max(k), which captures the best bound attainable by this parametric approach.

The authors conjecture that the incremental benefit of having k generalized boundary points, measured by g_max(k) = f_max(k) − f_max(0), is strictly subadditive in k. Intuitively, overlap between disks when ρ > 1 limits the additive gains from additional boundary points. A proof would formalize this limitation and sharpen the theoretical understanding of how boundary points contribute to exact covering guarantees in the parametric setting.