Conditioning random points by the number of vertices of their convex hull: the bi-pointed 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 ${\mathcal{H}}_{\lambda }$ 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 $\mathcal{C}$ in $ABC$ (incident with $A$ and $B$), ${\mathcal{H}}_{\lambda }$ is the unique curve maximizing the functional $\mathcal{C}\mapsto \mathsf{Area}(\mathcal{C}{)}^{\lambda }\mathsf{L}(\mathcal{C}{)}^3$ where $\mathsf{L}(\mathcal{C})$ is the affine perimeter of $\mathcal{C}$. We also give the logarithm expansion of the probability ${\mathbf{Q}}_{n,\lfloor n\lambda \rfloor }^{△\bullet \bullet }$, 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 ${\mathbf{Q}}_{n,m}^{\mathbf{K}}$ 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 ${\mathbf{Q}}_{n,m}^{\mathbf{K}}$, 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 $\lambda =0$.

57 pages