On splines and counting lattice points in polytopes

Let X be a (d × N)-matrix. We consider the variable polytope Π_X(u) = { w ≥ 0 : X w = u }. It is known that the function T_X that assigns to a parameter u ∈ R^d the volume of the polytope Π_X(u) is piecewise polynomial. Formulas of Khovanskii-Pukhlikov and Brion-Vergne imply that the number of lattice points in Π_X(u) can be obtained by applying a certain differential operator to the function T_X.
In this talk I will explain the facts mentioned above, prove a conjecture of Holtz and Ron on box splines and interpolation, and deduce an improved version of the Khovanskii-Pukhlikov formula. I will also mention some connections with matroid theory.
The talk will be based on arXiv:1211.1187 and arXiv:1305.2784.