We study the equivalent condition for the closure of any particular family of
cutting-planes to be polyhedral, from the perspective of convex geometry. We
also propose a new concept for valid inequalities of a convex set, namely the
finitely-irredundant inequality (FII), and show that a full-dimensional
cutting-plane closure is polyhedral, if and only if it has finitely many FIIs.
Based on those results we prove one of the problems left in Bodur et al.: the
k-aggregation closure of a covering set is a covering polyhedron