Free Discontinuities in Optimal Transport

Abstract

We prove a nonsmooth implicit function theorem applicable to the zero set of the difference of convex functions. This theorem is explicit and global: it gives a formula representing this zero set as a difference of convex functions which holds throughout the entire domain of the original functions. As applications, we prove results on the stability of singularities of envelopes of semi-convex functions, and solutions to optimal transport problems under appropriate perturbations, along with global structure theorems on certain discontinuities arising in optimal transport maps for the bilinear cost ${c(x, \bar{x}):=-\langle {x}, {\bar{x}}\rangle}$ for ${x,\bar{x} \in {\bf R}^n}$. For targets whose components satisfy additional convexity, separation, multiplicity, and affine independence assumptions, we show that these discontinuities occur on submanifolds of the appropriate codimension which are parameterized locally as differences of convex functions (DC, hence ${C^2}$ rectifiable), and—depending on the precise assumptions—${C^{1,\alpha}}$ smooth. Under these hypotheses, any ${n+1}$ affinely independent components of the target measure select at most one point from the source measure where the transport divides between all ${n+1}$ specified target components.