Adversarial examples have pointed out Deep Neural Networks vulnerability to\nsmall local noise. It has been shown that constraining their Lipschitz constant\nshould enhance robustness, but make them harder to learn with classical loss\nfunctions. We propose a new framework for binary classification, based on\noptimal transport, which integrates this Lipschitz constraint as a theoretical\nrequirement. We propose to learn 1-Lipschitz networks using a new loss that is\nan hinge regularized version of the Kantorovich-Rubinstein dual formulation for\nthe Wasserstein distance estimation. This loss function has a direct\ninterpretation in terms of adversarial robustness together with certifiable\nrobustness bound. We also prove that this hinge regularized version is still\nthe dual formulation of an optimal transportation problem, and has a solution.\nWe also establish several geometrical properties of this optimal solution, and\nextend the approach to multi-class problems. Experiments show that the proposed\napproach provides the expected guarantees in terms of robustness without any\nsignificant accuracy drop. The adversarial examples, on the proposed models,\nvisibly and meaningfully change the input providing an explanation for the\nclassification.\n