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.

22 pages

Cited by in corpus (4)

• Contact graphs of ball packings
• From canyons to valleys: Numerically continuing sticky hard sphere clusters to the landscapes of smoother potentials
• Diffusion dynamics for an infinite system of two-type spheres and the associated depletion effect
• Few distance sets in ${\ell }_p$ spaces and ${\ell }_p$ product spaces