  For $G$ a reductive group and $T\subset B$ a maximal torus and Borel
subgroup, Demazure modules are certain $B$-submodules, indexed by elements of
the Weyl group, of the finite irreducible representations of $G$. In order to
describe the $T$-weight spaces that appear in a Demazure module, we study the
convex hull of these weights - the Demazure polytope. We characterize these
polytopes both by vertices and by inequalities, and we use these results to
prove that Demazure characters are saturated, in the case that $G$ is simple of
classical Lie type. Specializing to $G=GL_n$, we recover results of Fink,
M\'esz\'aros, and St. Dizier, and separately Fan and Guo, on key polynomials,
originally conjectured by Monical, Tokcan, and Yong.
