  We study properties of convex hulls of (co)adjoint orbits of compact groups,
with applications to invariant theory and tensor product decompositions. The
notion of partial convex hulls is introduced and applied to define two
numerical invariants of a coadjoint orbit of a semisimple connected compact Lie
group. It is shown that the orbits, where any one of these invariants does not
exceed a given number $r$, form, upon intersection with a fixed Weyl chamber, a
rational convex polyhedral cone in that chamber, related to the
Littlewood-Richardson cone of the $r$-fold diagonal embedding of $K$. The
numerical invariants are shown to provide lower bounds for degrees of invariant
polynomials on irreducible unitary representations.
