Abstract: Convex functions naturally arise in many geometric contexts, such as optimal transport and computational geometry. The set of discontinuities of a convex function can be surprisingly intricate—potentially dense in its domain—yet it also exhibits some geometric structure, being (d-1)-rectifiable. In computational geometry, this set is related to the so called "medial axis" of a subset $K$ of the Euclidean space, i.e. the set of points which have more than one projection on $K$, which captures important topological and geometric information about $K$.

In this work, we establish new quantitative bounds on the size of the discontinuity set of the gradient of a convex, Lipschitz-continuous functions defined on bounded domains.  Our study is motivated by questions concerning the quantitative stability of the mapping that associates to a measure its pushforward under a fixed, possibly non-smooth, optimal transport map. Beyond this, our results have further applications in showing very fast convergence of Lloyd's algorithm for optimal uniform quantization.

For further information please contact elisur.magrini@unibocconi.it