Martin Winter: Wachspress Objects and the Reconstruction of Polytopes from Partial Metric Data

ABSTRACT: 

Wachspress objects are a family of constructions on convex polytopes that originate from geometric modelling and finite element analysis, but have recently been recognized as a much broader and surprising bridge between algebraic and convex geometry, linking also to statistics and mathematical physics. In the first part of the talk, I introduce the Wachspress coordinates via their many seemingly unrelated definitions. I then discuss a particular application: the reconstruction of convex polytopes from partial combinatorial and metric data. Questions of this nature have a long history and are intimately linked to both classical rigidity theory and real algebraic geometry. Even the following question turns out surprisingly hard: is a polytope uniquely determined by its edge-graph, edge lengths and the distance of each vertex from some interior point? If true, this would generalize and unify a number of known results, such as the Kirszbraun theorem and the reconstruction of matroids from their base exchange graph. I demonstrate how to approach and answer such questions using the Wachspress objects.