We consider the cones of divisors that are ample in codimension k on a variety of dimension n, for all possible values of k. Taking closures, for k=0 and k=n-1 we obtain the pseudoeffective and the nef cone respectively. For intermediate values of k, little is known about these cones. In this talk I will show that for Mori dream spaces (MDSs) all such cones are rational polyhedral. I will then discuss duality between such cones and cones of k-moving curves. For MDSs, a Weak Duality property holds, already proved by Payne and Choi. I will show that a Strong Duality property is satisfied by an interesting family of varieties. This is a joint work with O. Dumitrescu, C. Brambilla and L. Santana-Sánchez.