Asymptotic probability expansion for general compact convex sets

Establish the asymptotic logarithmic expansion of the probability that the convex hull of n+m independent uniform points in a compact convex planar set K has exactly n vertices, for fixed m/n tending to a positive parameter λ, under the stated hypothesis that the optimization functional over compact convex subsets of K has a unique maximizer C★.

Background

For a compact convex set K of area one, the paper denotes by Q^K_{n,m} the probability that the convex hull of n+m independent uniform points in K has n vertices. In the bi-pointed triangular case, the authors prove a logarithmic asymptotic expansion involving the parameter λ and the functional Area(C)^λ L(C)^3, where L is affine perimeter.

The analogous result for a general compact convex set K is stated only as a conjecture. The proposed asymptotic depends on the unique maximizer C★ of the functional PLK(C)=L(C)^3 Area(C)^λ, and the later discussion explains that the proof is obstructed by the lack of a sufficiently uniform asymptotic approximation for the triangular probabilities when the local ratios m_i/s_i are not uniformly bounded.