Polytopes with Polynomial Barycentric Coordinates and Low-Degree (Affine) Adjoints (joined Work with Martin Winter)

Given a convex polytope P, a system of "generalized barycentric coordinates" (GBCs for short) provides a canonical coordinate system intrinsic to the geometry of P. Many GBCs have been developed with applications in geometric modeling and finite element analysis.

In this talk, we study polytopes for which there exist polynomial GBCs or, more generally, rational GBCs of the form f_i/g, where g is a polynomial of low degree. We describe the geometric and combinatorial constraints imposed by low-degree denominators and provide a classification for polytopes with denominator degrees zero and one. Finally, we explain the relation to Wachspress coordinates, which are the rational GBCs that minimize the numerator degree.