  We consider an extended version of Horn's problem: given two orbits
$\mathcal{O}_\alpha$ and $\mathcal{O}_\beta$ of a linear representation of a
compact Lie group, let $A\in \mathcal{O}_\alpha$, $B\in \mathcal{O}_\beta$ be
independent and invariantly distributed random elements of the two orbits. The
problem is to describe the probability distribution of the orbit of the sum
$A+B$. We study in particular the familiar case of coadjoint orbits, and also
the orbits of self-adjoint real, complex and quaternionic matrices under the
conjugation actions of $\mathrm{SO}(n)$, $\mathrm{SU}(n)$ and $\mathrm{USp}(n)$
respectively. The probability density can be expressed in terms of a function
that we call the volume function. In this paper, (i) we relate this function to
the symplectic or Riemannian geometry of the orbits, depending on the case;
(ii) we discuss its non-analyticities and possible vanishing; (iii) in the
coadjoint case, we study its relation to tensor product multiplicities
(generalized Littlewood--Richardson coefficients) and show that it computes the
volume of a family of convex polytopes introduced by Berenstein and Zelevinsky.
These considerations are illustrated by a detailed study of the volume function
for the coadjoint orbits of $B_2=\mathfrak{so}(5)$.
