# Hall's Conjecture on Extremal Sets for Random Triangles

## TL;DR

This paper investigates Hall's conjecture on the probability that three random points form an acute triangle within convex regions, proving local maximality of the disk and reducing the conjecture to a finite computational problem.

## Contribution

The paper proves the disk is a local maximum for the probability in both 2D and 3D, and introduces new analytical tools to approach Hall's conjecture.

## Key findings

- Disk is a weak local maximum in 2D and 3D.
- Provides a local $C^{2,1/2}$ estimate on the probability.
- Reduces the conjecture to a finite computational verification.

## Abstract

In this paper we partially resolve Hall's conjecture about the distribution of random triangles. We consider the probability that three points chosen uniformly at random, in a bounded convex region of the plane, form an acute triangle. Hall's conjecture is the "isoprobabilistic inequality" which states that this probability should be maximized by the disk. We first prove that the disk is a weak local maximum for planar regions and that the ball is a weak local maximum in three dimensional regions. We then prove a local $C^{2,1/2}$-type estimate on the probability in the Hausdorff topology. This enables us to prove that the disk is a local maximum in the Gromov-Hausdorff topology (modulo congruences). Finally, we give an explicit upper bound on the isoperimetric ratio of the regions which maximize the probability and show how this reduces verifying the full conjecture to a finite, though currently intractable, calculation. An interesting aspect of our work is the use of tools from outside of geometric probability. We use an autocorrelation integral to provide the appropriate framework for the problem. When we study the problem in $\mathbb{R}^3$, we need non-Abelian harmonic analysis. To set up the proof that the disk is a strong local maximum, we use the Borsak-Ulam theorem.