Revisiting Gale Diagrams -
Studying the facial structure of polyhedra has been a vital part of research both from a combinatorial perspective as well as in convex optimization through linear programming. One tool developed for understanding the facial structure of polytopes is the Gale diagram. In joint work with Catherine Babecki, Venkat Chandrasekaran, and Amy Wiebe, we generalize Gale duality to encompass more general convex bodies. We are particularly interested in spectratopes, linear images of the spectraplex. The spectraplex is the analogue of the simplex in the cone of positive semidefinite matrices, the cone of interest in semidefinite programming. This work gives us a new combinatorial way to understand the facial structure of convex bodies beyond polytopes.