  We show quantitative stability results for the geometric "cells" arising in
semi-discrete optimal transport problems. Our results show two types of
stability, the first is stability of the associated Laguerre cells in measure,
without any connectedness or regularity assumptions on the source measure. The
second is stability in Hausdorff measure, under a Poincar{\`e}-Wirtinger
inequality and a regularity assumption equivalent to the Ma-Trudinger-Wang
conditions of regularity in Monge-Amp{\`e}re. This last result also yields
stability in the uniform norm of the dual potential functions, all three
stability results come with explicit quantitative bounds. Our methods utilize a
combination of graph theory, convex geometry, and Monge-Amp{\`e}re regularity
theory.
