AROUND SYLVESTER'S QUESTION IN THE PLANE. Issue 4 (16th August 2021) - Marckert, Jean‐François; Rahmani, Saïd
Abstract: 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 H n be the probability that the functions of Z i are the vertices of a convex polygon. Blaschke ( Ber. Verh. Sachs. Akad. Wiss. Leipzig Math.‐Phys . 69 (1917), 436–453) proved that Q T 4 ⩽ Q H 4 ⩽ Q D 4, where D is a disk and T a triangle. In the present paper we prove Q T 5 ⩽ Q H 5 ⩽ Q D 5 . One of the main ingredients of our approach is a new formula for Q H n of independent interest. We conjecture that the new formula we provide for Q H n will lead in the future to the complete proof that Q T n ⩽ Q T n ⩽ Q H n, for any n : we provide some partial results in this direction.