  Generalized alcoved polytopes are polytopes whose facet normals are roots in
a given root system. We call a set of points in an alcoved polytope a
generating set if there does not exist a strictly smaller alcoved polytope
containing it. The type $A$ alcoved polytopes are precisely the tropical
polytopes that are also convex in the usual sense. In this case the tropical
generators form a generating set. We show that for any root system other than
$F_4$, every alcoved polytope invariant under the natural Weyl group action has
a generating set of cardinality equal to the Coxeter number of the root system.
