  For an untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ with Cartan and
Borel subalgebras $\mathfrak{h} \subset \mathfrak{b} \subset \mathfrak{g}$,
affine Demazure modules are certain $U(\mathfrak{b})$-submodules of the
irreducible highest-weight representations of $\mathfrak{g}$. We introduce here
the associated affine Demazure weight polytopes, given by the convex hull of
the $\mathfrak{h}$-weights of such a module. Using methods of geometric
invariant theory, we determine inequalities which define these polytopes; these
inequalities come in three distinct flavors, specified by the standard,
opposite, or semi-infinite Bruhat orders. We also give a combinatorial
characterization of the vertices of these polytopes lying on an arbitrary face,
utilizing the more general class of twisted Bruhat orders.
