Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

Abstract

Abstract We show quantitative stability results for the geometric “cells” arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma–Trudinger–Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory.