Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results
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 B\'ar\'any (yet valid for a general convex domain $K$) and a result we initiated in the case where $K$ is a regular convex polygon.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2024
- arXiv:
- arXiv:2410.11706
- Bibcode:
- 2024arXiv241011706M
- Keywords:
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- Mathematics - Probability;
- Mathematics - Combinatorics
- E-Print:
- 21 pages, 19 figures