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Our goal is twofold. On one hand we show that the cones of divisors ample incodimension $k$ on a Mori dream space are rational polyhedral. On the otherhand we study the duality between such cones and the cones of $k$-moving curvesby means of the Mori chamber decomposition of the former. We give a new proofof the weak duality property (already proved by Payne and Choi) and we exhibitan interesting family of examples for which strong duality holds.

Duality and polyhedrality of cones for Mori dream spaces