In this article, we introduce a new algorithm for solving discrete optimal transport based on iterative resolutions of local versions of the dual linear program. We show a quantitative link between the complexity of this algorithm and the geometry of the underlying measures in the quadratic Euclidean case. This discrete method is then applied to investigate to wo optimal transport problems with geometric flavor: the regularity of optimal transport plan on oblate ellipsoids, and Alexandrov's problem of reconstructing a convex set from its Gaussian measure.