Around Sylvester's question in the plane

TL;DR

This paper investigates the probability that randomly chosen points form a convex polygon within a convex set, proving inequalities for five points and introducing a new formula that could extend these results to any number of points.

Contribution

The paper introduces a new formula for the probability of convex polygons from random points, advancing the understanding of geometric probability inequalities.

Findings

Proved inequalities for five points in convex sets.

Developed a new formula for the probability $Q^n_H$.

Showed Steiner symmetrization and shaking affect $Q^n_H$ in predictable ways.

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 $Q^n_H$ 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 $Q^n_H$ which permits to prove that Steiner symmetrization does not decrease $Q^5_H$, and that shaking does not increases it (this is the method Blaschke used in the $n=4$ case). We conjecture that the new formula we provide will lead in the future to the complete proof that $Q^n_T\leq Q^n_H\leq Q^n_D$ , for any $n$.