Spaces of Convex n-Partitions

Abstract

We construct and study the space ${\mathcal {C}}({\mathbb {R}}^d,n)$ of all partitions of ${\mathbb {R}}^d$ into n non-empty open convex regions (n-partitions). A representation on the upper hemisphere of an n-sphere is used to obtain a metric and thus a topology on this space. We show that the space of partitions into possibly empty regions ${\mathcal {C}}({\mathbb {R}}^d,\le n)$ yields a compactification with respect to this metric. We also describe faces and face lattices, combinatorial types, and adjacency graphs for n-partitions, and use these concepts to show that ${\mathcal {C}}({\mathbb {R}}^d,n)$ is a union of elementary semialgebraic sets.

The first author was funded by DFG through the Berlin Mathematical School. Research by the second author was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.