  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.
