Spaces of convex n-partitions

Abstract

We construct and study the space C(\R^d,n) of all partitions of \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 C(\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 C(\R^d,n) is a union of elementary semialgebraic sets.