Conditioning random points by the number of vertices of their convex hull: the bi-pointed case

TL;DR

This paper investigates the asymptotic shape of the convex hull boundary formed by random points in a triangle, revealing a phase transition between hyperbolic and parabolic limits, and extends results to general convex sets.

Contribution

It introduces a phase transition in the convex hull boundary shape conditioned on the number of vertices, and characterizes the limit shapes as solutions to an optimization problem.

Findings

Hyperbola as limit shape for linear vertex count ratio

Parabola as limit shape when one part is negligible

Explicit probability expansion for the hyperbola case

Abstract

Pick $N$ random points $U_1,\cdots,U_{N}$ independently and uniformly in a triangle ABC with area 1, and take the convex hull of the set $\{A,B,U_1,\cdots,U_{N}\}$. The boundary of this convex hull is a convex chain $V_0=B,V_1,\cdots,$ $V_{\mathbf{n}(N)}$, $V_{\mathbf{n}(N)+1}=A$ with random size $\mathbf{n}(N)$. The first aim of this paper is to study the asymptotic behavior of this chain, conditional on $\mathbf{n}(N)=n$, when both $n$ and $m=N-n$ go to $+\infty$. We prove a phase transition: if $m=\lfloor n\lambda\rfloor$ where $\lambda>0$, this chain converges in probability for the Hausdorff topology to an (explicit) hyperbola ${\cal H}_\lambda$ as $n\to+\infty$, 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 $B$), ${\cal H}_\lambda$ is the unique curve maximizing the functional ${\cal C}\mapsto {\sf Area}({\cal C})^{\lambda} {\sf L}({\cal C})^3$ where ${\sf L}({\cal C})$ is the affine perimeter of ${\cal C}$. We also give the logarithm expansion of the probability ${\bf Q}^{\triangle \bullet\bullet}_{n,\lfloor n\lambda\rfloor}$, that $\mathbf{n}(N)=n$ when $N=n+\lfloor n\lambda\rfloor$. Take a compact convex set $\mathbf{K}$ with area 1 in the plane, and denote by ${\bf Q}^{\mathbf{K}}_{n,m}$ the probability of the event that the convex hull of $n+m$ iid uniform points in $\mathbf{K}$ is a polygon with $n$ vertices. We provide some results and conjectures regarding the asymptotic logarithm expansion of ${\bf Q}^{\mathbf{K}}_{n,m}$, as well as results and conjectures concerning limit shape theorems, conditional on this event. These results and conjectures generalize B\'ar\'any's results, who treated the case $\lambda=0$.