Around Sylvester's question in the plane

Jean-Francois Marckert (1, 2) , Saïd Rahmani (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 CNRS - Centre National de la Recherche Scientifique

Pick $n$ points $Z_0,...,Z_{n-1}$ uniformly and independently at random in a compact convex set $H$ with non empty interior of the plane, and let $Q^n_H$ be the probability that the $Z_i$'s are the vertices of a convex polygon. Blaschke 1917 \cite{Bla} proved that $Q^4_T\leq Q^4_H\leq Q^4_D$, where $D$ is a disk and $T$ a triangle. In the present paper we prove $Q^5_T\leq Q^5_H\leq Q^5_D$. One of the main ingredients of our approach is a new formula for $Q^n_H$ of independent interest.\par We conjecture that the new formula we provide for $Q^n_H$ will lead in the future to the complete proof that $Q^n_T\leq Q^n_T\leq Q^n_H$ , for any $n$: we provide some partial results in this direction.

Domaines: Probabilités [math.PR]; Géométrie métrique [math.MG]

Jean-Francois Marckert, Saïd Rahmani. Around Sylvester's question in the plane. Mathematika, 2021, 67 (4), pp.860-884. ⟨hal-02913348⟩