Instability of martingale optimal transport in dimension d≥2

Stability of the value function and set minimizers w.r.t. given data is a desirable feature optimal transport problems. For classical Kantorovich problem, stability satisfied under mild assumptions in general frameworks such as one Polish spaces. However, for martingale problem several works based on different strategies established results Ronly. We show that restriction to dimension d=1 not accidental by presenting sequence marginal distributions R2 which neither stable nor minimizers. Our construction adapts any d≥2. d≥2 it also provides contradiction Wasserstein inequality Jourdain Margheriti d=1.