In this note, we prove that the disk is a local maximum for the geometric probability that three points chosen uniformly at random in a bounded convex region of the plane form an acute triangle. This provides progress towards a conjecture by Glen Hall, which states that the probability is maximized by the disk. We prove a corresponding result in three dimensions as well. We also prove that if the isoperimetric ratio is sufficiently large, that the probability of picking an acute triangle is small, and so the maximum must occur in some Gromov-Hausdorff compact set in the moduli space. 查看全文>>