  We study the relationship between the tensor product multiplicities of a
compact semisimple Lie algebra $\mathfrak{g}$ and a special function
$\mathcal{J}$ associated to $\mathfrak{g}$, called the volume function. The
volume function arises in connection with the randomized Horn's problem in
random matrix theory and has a related significance in symplectic geometry.
Building on box spline deconvolution formulae of Dahmen-Micchelli and De
Concini-Procesi-Vergne, we develop new techniques for computing the
multiplicities from $\mathcal{J}$, answering a question posed by Coquereaux and
Zuber. In particular, we derive an explicit algebraic formula for a large class
of Littlewood-Richardson coefficients in terms of $\mathcal{J}$. We also give
analogous results for weight multiplicities, and we show a number of further
identities relating the tensor product multiplicities, the volume function and
the box spline. To illustrate these ideas, we give new proofs of some known
theorems.
