Journées de Probabilités 2024

Probabilité que n points soient en position convexe dans un polygone quelconque : Résultats asymptotiques

13 juin 2024, 14:00

30m

Salle de Conférénces (Institut de mathématiques de Bordeaux)

Orateur

Ludovic Morin (LaBRI, Université de Bordeaux)

Description

The study of the probability that n points drawn uniformly and independently in a convex domain of area 1 (in the plane) are in convex position, meaning, they form the vertex set of a convex polygon, is quite an age-old question. The matter was risen at the end of the 19th century with Sylvester’s conjecture for $n=4$ points, solved by Blaschke in 1917. Since then, general results for $n$ points came one after the other in the square, the triangle or the disk, as well as other asymptotic results.
In this talk I will give an equivalent of the probability ${\mathbb{P}}_n$ that $n$ points are in convex position in a regular convex polygon to deduce an analogous result for any convex polygon; so far, the most precise formula was due to Bárány and identified the limit $n^2({\mathbb{P}}_n{)}^{1/n}$ (though Bárány’s formula holds for general convex domains).
Bárány also proved that a convex $n$-gon drawn uniformly in a fixed convex domain $K$ converges to a deterministic domain. Still working in the case where $K$ is a polygon, we present second order results for the fluctuations of the $n$-gon around this domain.