  In this short note we announce three formulas for the set of weights of
various classes of highest weight modules $\V$ with highest weight \lambda,
over a complex semisimple Lie algebra $\lie{g}$ with Cartan subalgebra
$\lie{h}$. These include, but are not restricted to, all (highest weight)
simple modules L(\lambda). We also assert that these formulas are the "best
possible", in that they do not hold in general for other highest weight modules
in a very precise sense. The proofs of the results in this note are included in
an updated copy (Version 3) of the paper arxiv:1301.1140 . The proofs involve
studying the convex hull of the set of $\lie{h}$-weights $\wt(\V)$ in their own
right. Thus, we show that if $\V = L(\lambda)$ is simple, or if \lambda\ is not
on a simple root hyperplane and $\V$ is arbitrary, the hull of the infinite set
$\wt(\V)$ is a convex polyhedron - i.e., cut out by only finitely many
hyperplanes. (This extends the notion of the Weyl polytope to arbitrary simple
modules L(\lambda).) It is also shown that the partially ordered set (under
quotienting) of modules $\V$ with fixed convex hull, has unique "largest" and
"smallest" elements.
