2006/08/31 by Hiroaki Terao
Engineering · Mathematics · #Arrow #Arrow's impossibility theorem #Calculus (dental) #Combinatorics #Computer science #Discrete mathematics #Hyperplane #Impossibility #Law #Mathematical economics #Mathematics #Mathematics and Applications #Point processes and geometric inequalities #Political science #Social choice theory #graph theory and CDMA systems #math.CO #msc:32S22 #msc:91B14
paper · pdf · doi:10.1016/j.aim.2007.02.006
published as Advances in Math. 214 (2007), 366-378
arxiv created 2007/02/10 · openalex publication_date 2007/03/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let \mathcal A be a nonempty real central arrangement of hyperplanes and \rm \bf Ch be the set of chambers of \mathcal A. Each hyperplane H defines a half-space H+ and the other half-space H-. Let B = \+, -\. For H∈ \mathcal A, define a map εH+ : \rm \bf Ch → B by εH+ (C)=+ (if C⊆ H+) and εH+ (C)= - (if C⊆ H-). Define εH-=-εH+. Let \rm \bf Chm = \rm \bf Ch×\rm \bf Ch×...×\rm \bf Ch (mtimes). Then the maps εH± induce the maps εH± : \rm \bf Chm → Bm . We will study the admissible maps Φ: \rm \bf Chm → \rm \bf Ch which are compatible with every εH±. Suppose |\mathcal A|≥ 3 and m≥ 2. Then we will show that \mathcal A is indecomposable if and only if every admissible map is a projection to a omponent. When \mathcal A is a braid arrangement, which is indecomposable, this result is equivalent to Arrow's impossibility theorem in economics. We also determine the set of admissible maps explicitly for every nonempty real central arrangement.