Adjoints of polytopes

In recent years the study of certain semi-algebraic sets known as "positive geometries" gained increasing attention due to its relevance in the sciences. Originally this concept was defined by physicists, but these sets also admit a beautiful structure when considering them from a mathematical viewpoint. The easiest class of examples consists of convex polytopes. The theory of positive geometries allows us to associate a polynomial known as the adjoint to each convex polytope. This polytope had several appearances in the study of polytopes in the past: Some examples include volumes of polar bodies and generalized barycentric coordinates of polytopes. In my talk I plan to present several different methods of computing the adjoint of a polytope from the perspective of positive geometry. These methods range from combinatorial formulas including triangulations to geometric interpolation problems.