Limit shape for conditioned convex hulls in general compact convex sets

Prove that, for a compact convex planar set K of area one and fixed λ>0, conditioning n+⌊nλ⌋ independent uniform points on their convex hull having exactly n vertices yields convergence in Hausdorff distance to the unique maximizer C★ of the functional C↦L(C)^3 Area(C)^λ, whenever that maximizer is unique.

Background

The paper proves the corresponding limit-shape theorem in the bi-pointed triangular model, where the limiting curve is an explicit hyperbola. For a general compact convex set K, deterministic geometric arguments show that every maximizer of the functional L(C)^3 Area(C)^λ has hyperbolic components between contact points with the boundary of K, and that the maximizer set is nonempty.

The probabilistic convergence statement remains unresolved. The authors explain that compactness gives subsequential limits, but proving a limit-shape theorem requires uniqueness of the accumulation point; the missing estimates are tied to controlling the optimization of the multi-parameter decomposition involving local vertex and interior-point counts.