Before revision: In the first part of this work , we uncover a connection between polynomial-reproducing bases of simplex splines and certain single-element liftings of oriented matroids associated to point configurations. We use this correspondence to construct a broad family of spline bases on generic point multisets, generalizing a known result on Delaunay configurations .
After revision: In the first part of this work , we uncover a connection between polynomial-reproducing spline spaces over subsets of conv(A) and fine zonotopal tilings of the zonotope Z(V) associated to the corresponding vector configuration. This link directly generalizes a known result on Delaunay configurations .