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

TL;DR

The work analyzes conditioning of random points by the number of convex-hull vertices in the unit triangle, focusing on the bi-pointed case with two distinguished triangle corners. It reveals a phase transition: when a linear fraction λ of the N points lies below the convex hull boundary (m ≈ λn), the hull boundary converges to an explicit hyperbola H_λ; when m = o(n), the limit recovers the classical parabola from Bárány et al. The approach blends exact finite-n decompositions, affine-invariance, and a variational principle for the maximizing curve via the functional Φ_λ(C)=L(C)^3 A(C)^λ, yielding an explicit optimization problem whose solution is the hyperbola parameterized by r_λ satisfying (sinh(2r_λ))/(2r_λ)=λ+1. The paper extends these bi-pointed insights to general compact convex sets K, formulating conjectures and partial results on the asymptotic expansion of Q^{K}_{n,m} and the corresponding limit shapes, with special treatment of regular κ-gons and disks; it also discusses algorithmic generation of the bi-pointed hull and situates the results within Bárány’s affine-length framework and Sylvester-type questions. Overall, it provides a rigorous link between combinatorial hull structure, deterministic geometry, and variational optimization, establishing a foundation for a general theory of limit shapes under convex-hull conditioning and outlining key conjectures for non-triangular domains.

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 $m=\lfloor nλ\rfloor$ where $λ>0$, this chain converges in probability for the Hausdorff topology to an (explicit) hyperbola ${\cal H}_λ$ 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 $B$), ${\cal H}_λ$ is the unique curve maximizing the functional ${\cal C}\mapsto {\sf Area}({\cal C})^λ {\sf L}({\cal C})^3$ where ${\sf L}({\cal C})$ is the affine perimeter of ${\cal C}$. We also give the logarithm expansion of the probability ${\bf Q}^{\triangle \bullet\bullet}_{n,\lfloor nλ\rfloor}$, that $\mathbf{n}(N)=n$ when $N=n+\lfloor nλ\rfloor$. 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 Bárány's results, who treated the case $λ=0$.

Figures (16)

• Figure 1: Representation of the unitary triangle $ABC$. On the first picture, 4 points $U_1,\cdots,U_4$ are represented. Together with $A$ and $B$, they form a convex chain. On the second picture, $12=4+8$ points $U_1,\cdots,U_{12}$ are taken uniformly and independently at random in $ABC$. The convex hull of $\{A,B, U_1,\cdots,U_{12}\}$ can be identified as a chain going from $A$ to $B$, passing here through four $U_i$, the other 8 being strictly below this chain.
• Figure 2: On this example, $N=12$, ${\bf n}(N)=6$ and ${\bf m}(N)=6$, $(K_6,K_5,K_4,K_3,K_2,K_1)=(3,0,2,1,0,0)$.
• Figure 3: Function $\lambda\mapsto \beta_\lambda$. We have $\beta_0=2+\log(2)$.
• Figure 4: Limiting curves for $\lambda$ going from $0.1$ to $5$ (by step $0.05$) and then from $5$ to $100$ (step of $1$). The curves become hot when $\lambda\to0$ and cold when $\lambda$ grows . On the left the parametric curve $\{(X(t),Y(t)), t\in[0,1]\}$ (which is thus the hyperbola ${\cal H}_{\lambda}$), in the center $t\mapsto X(t)$ and at the right $t\mapsto Y(t)$. When $\lambda$ is close to zero, $X$ is essentially linear, meaning that the abscissa of the points $V_i$ are somehow uniform on $[0,1]$, while a deformation arises as $\lambda$ grows. Observe that $X(t)$ goes from 2 to 0 because of our choice of taking the $(X_i)$ decreasing.
• Figure 5: ${\sf A}({\cal H}_{\lambda})$ as a function of $r_\lambda$

Theorems & Definitions (82)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Phase transition
• Remark 1.5
• Theorem 1.6
• Remark 1.7: Identification of the parabola ${\cal P}$ and the "hyperbola" ${\cal H}_0$
• Remark 1.8
• Remark 1.9
• Remark 1.10