Vector partition functions and their continuous analogues (multivariate splines) appear

in many different fields, including approximation theory (box splines and their discrete analogues),

symplectic geometry and representation theory (Duistermaat-Heckman measure and

weight multiplicity function/Kostant's partition function),

and discrete geometry (volumes and number of integer points of convex polytopes).

I will start by presenting the theory of the spaces spanned by the local pieces of these

piecewise (quasi-)polynomial functions and point out connections with matroid theory.

This theory has been developed in the 1980s by Dahmen and Micchelli. Later it has been

put in a broader context by De Concini, Procesi, Vergne and others.

Then I will present a refined version of the Khovanskii-Pukhlikov formula that relates the

volume and the number of integer points of a smooth lattice polytope.