Ludovic Morin’s seminar
The study of the probability that points drawn uniformly and independently in a convex domain of area 1 (in the plane) are in a convex position, i.e. form the set of vertices of a convex polygon, dates back to the end of the 19th century and Sylvester's conjecture for 4 points, which was solved by Blaschke in 1917. Since then, more general results for n points have followed for squares, triangles and circles, as well as other asymptotic results.
In this presentation I will give an equivalent of the probability Pn that n points are in a convex position in any regular convex polygon; until then the most precise formula, due to Bárány (which holds under more general conditions, however), identified the limit n2 Pn1/n.
Bárány also proved that a uniform convex polygon drawn in a convex K converges to a deterministic domain. Still in the case of regular polygons, we present some 2nd order results concerning the fluctuations of the convex envelope around this domain.