What is the earliest published statement and proof of the well-known result: for a simple Lie algebra over $\mathbb{C}$ or other algebraically closed field of characteristic 0, the convex hull (in the dual of a fixed Cartan subalgebra) of the set of weights in a finite dimensional irreducible representation of highest weight $\lambda$ without the weight 0 contains only the weights $w \lambda$ (with $w$ in the Weyl group) iff these are the sole weights of the representation (in which case $\lambda$ is usually called "minuscule").

I've always tended to think of this result as being due to Kostant---somewhere in his early papers---or maybe Bourbaki. But this is too vague.

[Sorry to have omitted the needed assumption on the weight 0 in the earlier version; this amounts to requiring that $\lambda$ does not lie in the root lattice.]