  Let $\Phi$ be a finite crystallographic irreducible root system and $\mathcal
P_{\Phi}$ be the convex hull of the roots in $\Phi$. We give a uniform explicit
description of the polytope $\mathcal P_{\Phi}$, analyze the
algebraic-combinatorial structure of its faces, and provide connections with
the Borel subalgebra of the associated Lie algebra. We also give several
enumerative results.
