The expectation maximization (EM) algorithm is a popular method of density estimation for Gaussian mixture models. Fundamental to understanding the performance of this algorithm is to understand the set of points closest to a given Gaussian, where “closest" is defined in terms of the maximum likelihood function. We call this set of points a Gaussian Voronoi cell. In this talk, I will outline new results regarding the geometry and combinatorics of Gaussian Voronoi cells. This is joint work with Joe Kileel.