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Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers

Authors Elizaveta Streltsova , Uli Wagner

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Discrete mathematics
Keywords
  • Levels in arrangements
  • k-sets
  • k-facets
  • convex polytopes
  • f-vector
  • h-vector
  • g-vector
  • Dehn-Sommerville relations
  • Radon partitions
  • Gale duality
  • g-matrix

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Abstract

A long-standing conjecture of Eckhoff, Linhart, and Welzl, which would generalize McMullen’s Upper Bound Theorem for polytopes and refine asymptotic bounds due to Clarkson, asserts that for k ⩽ ⌊(n-d-2)/2⌋, the complexity of the (⩽ k)-level in a simple arrangement of n hemispheres in S^d is maximized for arrangements that are polar duals of neighborly d-polytopes. We prove this conjecture in the case n = d+4. By Gale duality, this implies the following result about crossing numbers: In every spherical arc drawing of K_n in S² (given by a set V ⊂ S² of n unit vectors connected by spherical arcs), the number of crossings is at least 1/4 ⌊n/2⌋ ⌊(n-1)/2⌋ ⌊(n-2)/2⌋ ⌊(n-3)/2⌋. This lower bound is attained if every open linear halfspace contains at least ⌊(n-2)/2⌋ of the vectors in V.
Moreover, we determine the space of all linear and affine relations that hold between the face numbers of levels in simple arrangements of n hemispheres in S^d. This completes a long line of research on such relations, answers a question posed by Andrzejak and Welzl in 2003, and generalizes the classical fact that the Dehn-Sommerville relations generate all linear relations between the face numbers of simple polytopes (which correspond to the 0-level).
To prove these results, we introduce the notion of the g-matrix, which encodes the face numbers of levels in an arrangement and generalizes the classical g-vector of a polytope.

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Elizaveta Streltsova and Uli Wagner. Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 75:1-75:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.75

Author Details

Elizaveta Streltsova
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Uli Wagner
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria

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