Moment polytopes on Sasaki manifolds and volume minimization

We show that transverse coupled K\"ahler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the K\"ahler cone, and the non-coupled case leads to a known existence result of a transverse K\"ahler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled K\"ahler-Einstein metrics.