Interpolation, Box Splines, and Lattice Points in Zonotopes

Let X be a totally unimodular list of vectors in some lattice. Let BX be the box spline defined by X. Its support is the zonotope Z(X). We show that any real-valued function defined on the set of lattice points in the interior of Z(X) can be extended to a function on Z(X) of the form p(D)BX in a unique way, where p(D) is a differential operator that is contained in the so-called internal $\mathcal P$-space. This was conjectured by Olga Holtz and Amos Ron. We also point out connections between this interpolation problem and matroid theory, including a deletion-contraction decomposition.