Longest convex chains

TL;DR

This paper studies the properties of the longest convex chains formed by random points in a triangle, establishing their expected length, concentration around the mean, and limit shape.

Contribution

It determines the order of magnitude of the expected length of the longest convex chain and proves concentration and limit shape results.

Findings

Expected length of the longest convex chain is characterized.

Longest convex chains are highly concentrated around their mean.

A limit shape for the longest convex chains is established.

Abstract

Assume $X_n$ is a random sample of $n$ uniform, independent points from a triangle $T$. The longest convex chain, $Y$, of $X_n$ is defined naturally. The length $|Y|$ of $Y$ is a random variable, denoted by $L_n$. In this article, we determine the order of magnitude of the expectation of $L_n$. We show further that $L_n$ is highly concentrated around its mean, and that the longest convex chains have a limit shape.