Projective geometry of Wachspress coordinates

Algebra Seminar
Tuesday, April 16, 2019 - 12:00pm for 1 hour (actually 50 minutes)
Skiles 005
KathlĂ©n Kohn – ICERM and University of Oslo –
Greg Blekherman
Wachspress defined barycentric coordinates on polygons in 1975. Warren generalized his construction to higher dimensional polytopes in 1996. In contrast to the classical case of simplices, barycentric coordinates on other polytopes are not unique. So the coordinates defined by Warren are now commonly known as Wachspress coordinates. They are used in a variety of applications, such as geometric modeling.
We connect the constructions by Warren and Wachspress by proving the conjecture that there is a unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. This is the adjoint polynomial introduced by Warren. Our formulation is the natural generalization of Wachspress' original idea.
The algebraic geometry of the map defined by the Wachspress coordinates was studied in the case of polygons by Irving and Schenk in 2014. We extend their results to higher dimensional polytopes. In particular, we show that the image of this Wachspress map is the projection from the image of the adjoint. For three-dimensional polytopes, we show that their adjoints are adjoints of K3- or elliptic surfaces. This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.