Illumination complexes, Δ-zonotopes, and the polyhedral curtain theorem

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−ΔΔ−Δ of a simplex Δ  , or the dual of the convex hull of the root system AnAn. 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’.