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 ${\cal H}\lambda$ 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 $\lambda=0$.