2025/10/30 by Jean‐François Marckert, Marckert, Jean-François, Ludovic Morin +1
Mathematics · Computer Science · #Point processes and geometric inequalities #Computational Geometry and Mesh Generation #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2510.26330
Pick N random points U1,⋯,UN independently and uniformly in a triangle ABC with area 1, and take the convex hull of the set \A,B,U1,⋯,UN\. The boundary of this convex hull is a convex chain V0=B,V1,⋯, Vn(N), Vn(N)+1=A with random size n(N). The first aim of this paper is to study the asymptotic behavior of this chain, conditional on n(N)=n, when both n and m=N-n go to +∞. We prove a phase transition: if m=\lfloor nλ\rfloor where λ>0, this chain converges in probability for the Hausdorff topology to an (explicit) hyperbola \cal Hλ as n→+∞, while, if m=o(n), the limit shape is a parabola. We prove that this hyperbola is solution to an optimization problem: among all concave curves \cal C in ABC (incident with A and B), \cal Hλ is the unique curve maximizing the functional \cal C↦ \sf Area(\cal C)λ \sf L(\cal C)3 where \sf L(\cal C) is the affine perimeter of \cal C. We also give the logarithm expansion of the probability \bf Q\triangle \bullet\bulletn,\lfloor nλ\rfloor, that n(N)=n when N=n+\lfloor nλ\rfloor. Take a compact convex set K with area 1 in the plane, and denote by \bf QKn,m the probability of the event that the convex hull of n+m iid uniform points in K is a polygon with n vertices. We provide some results and conjectures regarding the asymptotic logarithm expansion of \bf QKn,m, as well as results and conjectures concerning limit shape theorems, conditional on this event. These results and conjectures generalize Bárány's results, who treated the case λ=0.