Illumination complexes, {\Delta}-zonotopes, and the polyhedral curtain theorem

TL;DR

This paper introduces illumination complexes based on { Delta}-zonotopes, demonstrating their use as configuration spaces to prove new fair division theorems, including the polyhedral curtain theorem, related to classical ham sandwich and splitting necklace theorems.

Contribution

It develops the concept of illumination complexes from { Delta}-zonotopes and applies them to establish new fair division results, notably the polyhedral curtain theorem.

Findings

Introduction of illumination complexes based on { Delta}-zonotopes.

Application of these complexes to prove new fair division theorems.

Establishment of the polyhedral curtain theorem as a relative of classical theorems.

Abstract

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 {\Delta}-zonotope (generalized rhombic dodecahedron), known also as the dual of the difference body {\Delta} - {\Delta} of a simplex {\Delta}, or the dual of the convex hull of the root system A_n. 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'.