Dimension and structure of higher-order Voronoi cells on discrete sites

We study the structure of higher-order Voronoi cells on a discrete set of sites in $\mathbb{R}^n$, focussing on the relations between cells of different order, and paying special attention to the ill-posed case when a large number of points lie on a sphere. In particular, we prove that higher order cells of dimension $n-1$ do not exist, even though high-order Voronoi cells may have empty interior. We also present a number of open questions.