What does this research mean for the field?

The exact asymptotic behavior of the probability that n uniformly and independently distributed points in a general compact convex polygon form a convex position is determined as n approaches infinity. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The research aims to determine the asymptotic behavior of the probability that randomly selected points in a convex polygon are in convex position.

May 28, 2026

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

Key Points

The research aims to determine the asymptotic behavior of the probability that randomly selected points in a convex polygon are in convex position.
Analyzed the probability P_K(n) as n approaches infinity.
Improved upon existing results for convex sets.
Built on previous research in regular convex polygons.
Established the exact asymptotic behavior of P_K(n) as n tends to infinity.
Provided insights that enhance the understanding of points in convex position.
Validated results for general convex polygons beyond regular cases.

Abstract

Abstract Let PK (n) be the probability that n points z₁, , zₙ picked uniformly and independently in K, a compact convex polygon in R² with non-empty interior, are in convex position, that is, form the vertex set of a convex polygon. In this paper, the exact asymptotic behaviour of PK (n) is determined as n+. This improves on a famous result of Bárány 1999 Ann. Probab. 27, 2020–2034 (yet valid for a general convex set K) and a result initiated in the case where K is a regular convex polygon (Morin 2025 Adv. Appl. Probab. 57, 811–870).

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