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.