Lundi 15 septembre : Ludovic Morin

Title : The magic triangle & other stories

Summary : Let P_K(n,m) be the probability that the convex hull of n+m points drawn uniformly and independently in a convex set K of area 1 (in the plane) has exactly n vertices.
Initial works on this quantity go back to the end of the 19th century and Sylvester's four points problem that focuses on the set K that optimizes the probability P_K(4,0). It was solved by Blaschke in 1917, and since then, more general results have followed for P_K(n,0) when K is a parallelogram, a triangle or a circle, as well as other asymptotic results.
In particular, an important work of 2000 by Bárány et al. gives the exact probability that n i.i.d. uniform points in a triangle abc form a convex chain between a and b (\ie the n points are on the boundary of the convex hull of these n points together with a and b). Furthermore, the limit of such a convex chain as n\to+\infty is the parabola joining a to b that is tangent to both ac and cb.
In the first part of this presentation, after introducing this tool and its properties, I'll try to demonstrate how it relates to P_K(n,0).
In the second part of the talk, I will study a generalized version of this magic triangle : n+m uniform points are drawn in abc and we look at the probability that exactly n points among the n+m are on the boundary of the convex hull of these points together with a and b.
We give asymptotics of the probability of such an event and formulate convergence theorems for this convex hull for special values of m. Based on these recent results, we'll conclude this talk with a few conjectures on the study of P_K(n,m).
Joint work with Jean-François Marckert.


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