The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we introduce the notion of strong c-concavity, and we show that it plays an important role for proving stability results in optimal transport for general cost functions c. We then introduce a differential criterion for proving that a function is strongly c-concave, under an hypothesis on the cost introduced originally by Ma-Trudinger-Wang for establishing regularity of optimal transport maps. Finally, we provide two examples where this stability result can be applied, for cost functions taking value +$\infty$ on the sphere: the reflector problem and the Gaussian curvature measure prescription problem.

翻译：从数学角度看,在措施变化的情况下,稳定最佳运输问题的解决办法至关重要:它与数字方法的趋同密切相关,以解决最佳运输问题,并证明许多最佳运输应用的合理性;在本条中,我们引入了强大的c-cocavity概念,我们表明,该概念在证明稳定使一般成本功能的运输达到最佳效果方面起着重要作用;c。 然后,我们引入了一个差别标准,根据最初由Ma-Trudinger-Wang为确定最佳运输图的规律性而提出的费用假设,证明一个功能十分复杂;最后,我们提供了两个例子,说明可以应用这种稳定结果,用于成本功能,即成本值+$/infty$在现场:反射器问题和高斯曲线测量测量法时效问题。