Partial convex hulls of coadjoint orbits and degrees of invariants

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.