The magic triangle & other stories

by Ludovic Morin (Université de Bordeaux)

Tuesday Jun 17, 2025, 9:50 AM → 10:50 AM Europe/Paris

Salle F. Pellos (1R2-207)

Let P_K(n,m) be the probability that the convex hull of n+m points drawn uniformly and independently in a convex domain K of area 1 (in the plane) has exactly n vertices.
It all started with the study of the probability P_K(4,0), that goes back to the end of the 19th century and Sylvester’s four points problem, which was solved by Blaschke in 1917. 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, a surprising result by Bárány et al. in 2000 gives the exact probability that n i.i.d. uniform points in a triangle form a convex chain between two chosen vertices (i.e. the boundary of the convex hull of the n points together with two vertices of the triangle contains the n points).
In the first part of this presentation, after introducing this tool and its properties, I’ll try to demonstrate how a substantial number of the results that have emerged around P_K(n,q) can be seen as a direct or indirect consequence of this so-called “magic” triangle. In the second part of the talk, I will study a generalized version of this magic triangle, that allowed us to reach some recent results for P_K(n,m).