[Paper Review] Illumination complexes, {\Delta}-zonotopes, and the polyhedral curtain theorem

Rade T. Živaljević|arXiv (Cornell University)|Jul 19, 2013

Point processes and geometric inequalities11 references3 citations

TL;DR

This paper introduces illumination complexes and ∆-zonotopes as configuration spaces to prove a new fair division theorem—the polyhedral curtain theorem—establishing that d continuous measures in ℝᵈ can be simultaneously bisected by a ∆-curtain, a conical polyhedral hypersurface derived from a simplex. The result generalizes the ham sandwich and splitting necklace theorems with a combinatorial, geometric approach using configuration spaces and topological methods like the Borsuk-Ulam theorem.

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'.

Motivation & Objective

• To develop a new class of configuration spaces—illumination complexes and ∆-zonotopes—for studying fair division problems in geometric measure theory.
• To generalize the ham sandwich and splitting necklace theorems by introducing the polyhedral curtain theorem as a combinatorial alternative to hyperplane bisectors.
• To establish a topological framework using configuration spaces homeomorphic to spheres to prove the existence of fair divisions with prescribed combinatorial structure.
• To extend the theory to polynomial splines and curved fair division, showing that ∆-generated splines can bisect multiple measures in ℝ² via paraboloid embeddings.

Proposed method

• Constructs configuration spaces A(Q, F, S) by gluing copies of a convex body Q (e.g., a hexagon or ∆-zonotope) along boundaries according to a fan structure F and party labels S.
• Uses the Borsuk-Ulam theorem on the configuration space A(R∆, F∆, [2]) ≅ ∂(♦d) ≅ Sd−1 to prove existence of a fair division by a ∆-curtain.
• Applies a limiting and compactness argument to extend results from finite partitions to continuous measures, showing that ∆-curtains can approximate hyperplane bisectors.
• Employs the Veronese-type embedding W: ℝ² → ℝ³ defined by (x, y) ↦ (x, y, x² + y²) to lift planar measures to a paraboloid, enabling construction of circular splines that bisect measures.
• Defines ∆-generated splines as piecewise circular curves derived from the face directions of a simplex ∆⊂ℝ³, which control the slopes and combinatorics of the curtain.
• Uses the join decomposition of complementary faces of a simplex to define (d−2)-dimensional polyhedral spheres Sd−2θ, which generate the conical structure of ∆-curtains.

Experimental results

Research questions

• RQ1 Can a single polyhedral hypersurface (a ∆-curtain) simultaneously bisect d continuous measures in ℝᵈ, analogous to the ham sandwich theorem?
• RQ2 How can the configuration space of fair divisions be modeled using illumination complexes and ∆-zonotopes?
• RQ3 To what extent can the combinatorial structure of a simplex ∆⊂ℝᵈ be used to prescribe the slopes and topology of a fair dividing surface?
• RQ4 Can the polyhedral curtain theorem be extended to curved fair division, such as circular splines in ℝ²?
• RQ5 What is the relationship between the polyhedral curtain theorem and the multidimensional splitting necklace theorem?

Key findings

• The polyhedral curtain theorem (Theorem 3) establishes that for any d continuous measures in ℝᵈ, there exists a ∆-curtain H = x + cone(Sd−2θ) that bisects each measure equally, i.e., µj(H+) = µj(H−) for all j = 1, ..., d.
• The configuration space A(Q, F, [2]) for two-party fair division of a hexagon Q is homeomorphic to the 2-sphere S², providing a topological foundation for the Borsuk-Ulam argument.
• The theorem is proven via the Borsuk-Ulam theorem applied to the configuration space A(R∆, F∆, [2]) ≅ Sd−1, which encodes all possible divisions and allocations.
• The existence of fair divisions with prescribed combinatorial structure (e.g., cone directions from a simplex) is guaranteed by the topological structure of the configuration space.
• For polynomial splines, the paper shows that ∆-generated circular splines in ℝ² can bisect three measurable sets simultaneously, with the number and shape of the spline controlled by the face structure of a tetrahedron ∆⊂ℝ³.
• The limiting argument in Proposition 24 shows that the ∆-curtain theorem implies the more general Theorem 21 for q-partitions, by approximating cone structures from large simplices.