Spaces of convex \(n\)-partitions (Q2417587)

A convex \(n\)-partition of \(\mathbb{R}^d\) is a collection of \(n\) mutually disjoint, nonempty open convex regions such that union of their closures is \(\mathbb{R}^d\). The authors study the space \(\mathcal{C}(\mathbb{R}^d,n)\) of all convex \(n\)-partitions of \(\mathbb{R}^d\). Representations of convex \(n\)-partitions of \(\mathbb{R}^d\) as convex \(n\)-partitions of the unit \(d\)-sphere are used to introduce a metric on \(\mathcal{C}(\mathbb{R}^d,n)\) and show that the analogously defined topological space with empty regions allowed is a natural compactification of \(\mathcal{C}(\mathbb{R}^d,n)\). Also investigated are face structures and combinatorial types of partitions, as well as realization spaces for partitions of a given combinatorial type. It is shown that \(\mathcal{C}(\mathbb{R}^d,n)\) can be described as a finite union of semi-algebraic sets. For the entire collection see [Zbl 1411.52002].