The Optimal transport (OT) problem and its associated Wasserstein distance have recently become a topic of great interest in the machine learning community. However, its underlying optimization problem is known to have two major restrictions: (i) it strongly depends on the choice of the cost function and (ii) its sample complexity scales exponentially with the dimension. In this paper, we propose a general formulation of a minimax OT problem that can tackle these limitations by jointly optimizing the cost matrix and the transport plan, allowing us to define a robust distance between distributions. We propose to use a cutting-set method to solve this general problem and show its links and advantages compared to other existing minimax OT approaches. Additionally, we use this method to define a notion of stability allowing us to select the ground metric robust to bounded perturbations. Finally, we provide an experimental study highlighting the efficiency of our approach.