AROUND SYLVESTER'S QUESTION IN THE PLANE

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...