Our goal is twofold. On one hand, we show that the cones of divisors ample in codimension
k on a Mori dream space are rational polyhedral. On the other hand, we study the duality
between such cones and the cones of k-moving curves by means of the Mori chamber decomposition
of the former.We give a new proof of the weak duality property (already proved by
Payne and Choi) and we exhibit an interesting family of examples for which strong duality
holds.
