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Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results

Ludovic Morin

Abstract

Let $\mathbb{P}_K(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $K$, a non-flat compact convex polygon in $\mathbb{R}^2$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\mathbb{P}_K(n)$ when $n\to\infty$. This improves on a famous result of Bárány (yet valid for a general convex domain $K$) and a result we initiated in the case where $K$ is a regular convex polygon.

Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results

Abstract

Let be the probability that points picked uniformly and independently in , a non-flat compact convex polygon in , are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of when . This improves on a famous result of Bárány (yet valid for a general convex domain ) and a result we initiated in the case where is a regular convex polygon.

Paper Structure

This paper contains 17 sections, 15 theorems, 68 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notation.
  3. The analysis is simpler when $K\in\mathbf{P}^{\mathcal{T}}$:
  4. What happens when $K$ is not in $\mathbf{P}^{\mathcal{T}}$ ?
  5. A quick history.
  6. Notation.
  7. Definition of the $\mathsf{PCP}$ (\ref{['fig12']}).
  8. "Contact points" (see \ref{['fig:super']}).
  9. A result on tangent limit shapes and polygons
  10. The joint distribution of $(\boldsymbol\ell^{(n)}[\kappa],\mathbf{s}^{(n)}[\kappa])$
  11. Notation.
  12. Notation.
  13. The quantity $\mathop{\mathrm{AP}}\nolimits^*(K)$ in $f^{(n)}_K$
  14. General case
  15. Notation.
  16. ...and 2 more sections

Key Result

Theorem 1.1

barany2 For any compact convex set $G$ with non empty interior, where and $\mathop{\mathrm{AP}}\nolimits(S)$ denotes the affine perimeter of a convex set $S$.

Figures (10)

  • Figure 1: An example of $K\in\mathbf{P}_6$
  • Figure 2: The set of points in red are $n=200$ uniform points conditioned to be in convex position. The boundary of their convex hull is very close to a green curve being the boundary of the domain ${\sf Dom}(\pentagon)$.
  • Figure 3: Two examples of convex polygons drawn in blue, and their limit shape drawn in red. On the left, $K_L$ is in $\mathbf{P}_5^{\mathcal{T}}$, for its limit shape is tangent to every side. This is not the case anymore on the right: $K_R\in \mathbf{P}_7$, but ${\sf Tangency}_7(K_R)=\{1,2,3,5,6\}$ so that $K_R\notin\mathbf{P}_7^{\mathcal{T}}$.
  • Figure 4: If $K$ belongs to $\mathbf{P}_{\kappa}^{\mathcal{T}},$ the boundary of ${\sf Dom}(K)$ is tangent to every side of $K$ at ${\sf p}_j$ for the $j^{th}$ side. The triangles hached with bricks patterns are the triangle $T_i$, $i\in\{1,\ldots,\kappa\}$.
  • Figure 6: Two example of $\mathsf{PCP}$ for some $K\in\mathbf{P}_6$ (drawn in black). The $\mathsf{PCP}({z}[n])$ is drawn in blue, its side-lengths $c[\kappa]$ are represented in blue as well, and the side-distances $\ell[\kappa]$ are drawn in red (on the left only). On the left picture, all side-lengths $c[\kappa]$ are nonzero whereas it is not the case anymore on the right picture.
  • ...and 5 more figures

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 1.8
  • Theorem 2.1: Limit shape theorem, Báránybarany1
  • Lemma 2.2
  • ...and 16 more