Authors: Živaljević, Rade 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Illumination complexes, Δ-zonotopes, and the polyhedral curtain theorem
Journal: Computational Geometry: Theory and Applications
Volume: 48
Issue: 3
First page: 225
Last page: 236
Issue Date: 1-Jan-2015
Rank: M22
ISSN: 0925-7721
DOI: 10.1016/j.comgeo.2014.10.003
Illumination complexes are examples of 'flat polyhedral complexes' which arise if several copies of a convex polyhedron (convex body) Q are glued together along some of their common faces (closed convex subsets of their boundaries). A particularly nice example arises if Q is a Δ-zonotope (generalized rhombic dodecahedron), known also as the dual of the difference body Δ-Δ of a simplex Δ, or the dual of the convex hull of the root system An. We demonstrate that the illumination complexes and their relatives can be used as 'configuration spaces', leading to new 'fair division theorems'. Among the central new results is the 'polyhedral curtain theorem' (Theorem 3) which is a relative of both the 'ham sandwich theorem' and the 'splitting necklaces theorem'.
Keywords: Fair division | Illumination complexes | Splitting necklaces | Zonotope
Publisher: Elsevier
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Topology, geometry and global analysis on manifolds and discrete structures 

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