Kuperberg $S{L}_3$ webs from convex sets in the affine building

In 1997 Kuperberg gave a presentation for the spider categories of the rank 2 Lie algebras and showed that the non-elliptic webs form a basis in each invariant space. Fontaine, Kamnitzer, Kuperberg then showed that in type $A_2$ the non-elliptic webs are dual to CAT(0) diskoids in the affine building. Each such CAT(0) diskoid, aka dual non-elliptic web, is a triangulation of a generic polygon $P$ in the building. In this talk, we discuss work in progress for characterizing the dual non-elliptic webs as the intersection of two convex hulls of $P$ in the affine building. The convex sets are related to tropical convexity as discussed in work by Joswig, Sturmfels, Yu, and by Leon Zhang. We hope that these convexity ideas can be applied to higher rank $A_n$, where a good basis of webs is not known.