Longest convex chains with i.i.d. points

Abstract

Sample $n$ i.i.d. points from a triangle, according to some bounded density function. Given two vertices $A,B$ of the triangle, what is the maximum number of samples that form a convex chain with initial point $A$ and terminal point $B$? We show that to leading order, the answer is $cn^{1/3}$, generalizing a result of Ambrus and B\'ar\'any that considered uniformly distributed points. Furthermore, we express the constant $c$ using a variational formula whose maximizer (if unique) gives the limiting curve formed by the longest convex chain. By comparison, for $n$ i.i.d. samples from the unit square, the length of the longest monotone chain is asymptotically $c'n^{1/2}$. Despite the difference in scale, our formula is nicely connected to one established for $c'$ by Deuschel and Zeitouni.