  The necessary and sufficient Horn inequalities which determine the
non-vanishing Littlewood-Richardson coefficients in the cohomology of a
Grassmannian are recursive in that they are naturally indexed by non-vanishing
Littlewood-Richardson coefficients on smaller Grassmannians. We show how
non-vanishing in the Schubert calculus for cominuscule flag varieties is
similarly recursive. For these varieties, the non-vanishing of products of
Schubert classes is controlled by the non-vanishing products on smaller
cominuscule flag varieties. In particular, we show that the lists of Schubert
classes whose product is non-zero naturally correspond to the integer points in
the feasibility polytope, which is defined by inequalities coming from
non-vanishing products of Schubert classes on smaller cominuscule flag
varieties. While the Grassmannian is cominuscule, our necessary and sufficient
inequalities are different than the classical Horn inequalities.
