Hall's Conjecture

Hall's conjecture was originally put forward by my undergraduate advisor, Glen Hall, in his paper Acute Triangles in the n-Ball. While trying to prove the conjecture, I generated a lot of pictures and calculations. Here are some of the files that I found helpful. The results from these figures were used in my paper Hall's Conjecture on Extremal Sets for Random Triangles.

  • This Mathematica notebook contains the calculation of a lemma on Legendre polynomials that was used to understand the problem in 3 dimensions.

    Hall's calculations in 3D

For this project, it was difficult for me to visualize the geometry. Geogebra was a crucial tool, because it allowed me to manipulate sketches to see how small deformations affect the geometry.

  • These two Geogebra file contain a disk and the region of convex triangles given two fixed points. The first is related to the "autocorrelation term" while the second can be modified more freely.

    Hall's Problem

    Hall's Figure 2

  • The "Canonical Homotopy" Geogebra figured helped me understand how small translations and small dilations of convex regions interact. This was needed for Lemma 12, which constructed a canonical homotopy.

    Canonical Homotopy

  • This Geogebra figure was useful in calculating how the "autocorrelation term" changes as one deforms a convex region. It was useful for Lemma 14.

    Strong Maximum Autocorrelation

  • This Geogebra figure helped me visualize how the locus of right triangles can intersect the boundary of a convex region. This figure was helpful for Lemma 15.

    Strong Maximum Right Triangles

  • This Geogebra figure helped me picture how the volume of right triangles changes as the boundary of a convex region changes. This figure was also helpful for Lemma 15.

    Strong Maximum Right Triangles 2