Spaces of convex n-partitions

TL;DR

This paper introduces a topological and combinatorial framework for the space of convex n-partitions of Euclidean space, including metrics, compactifications, and structural descriptions of faces and adjacency.

Contribution

It constructs a new topological space for convex n-partitions, providing metrics, compactifications, and combinatorial characterizations that advance understanding of partition structures.

Findings

The space of convex n-partitions is metrizable and compactified.

Faces and face lattices are characterized and related to semialgebraic sets.

The space decomposes into elementary semialgebraic components.

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.