Spaces of Convex n-Partitions

  • Emerson LeónEmail author
  • Günter M. Ziegler
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)


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.



This paper presents main results of the doctoral thesis of the first author [10]. We are very grateful to both referees for very valuable and thoughtful comments.


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© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Depto. de MatemáticasUniversidad de los AndesBogotáColombia
  2. 2.Inst. Mathematics, FU BerlinBerlinGermany

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