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Sublevels in arrangements and the spherical arc crossing number of complete graphs

Elizaveta Streltsova, Uli Wagner

TL;DR

The paper develops a unified framework for sublevels in 3D arrangements via the g-matrix, establishing nonnegativity results that imply the Eckhoff–Linhart–Welzl conjecture for $d=n-4$. It then connects these combinatorial bounds to spherical arc drawings of $K_n$, proving a universal lower bound on the crossing number $cr(V)$ that matches Hill's conjectured value $X(n)$ for coneighborly configurations. Gale duality ties $f^*$-bounds to sublevel bounds, yielding exact upper bounds in rank $3$ and clarifying when equality is achieved. Overall, the work links generalized upper bound phenomena, mutation-driven changes, and geometric dualities to derive precise combinatorial constraints and optimal configuration structures.

Abstract

Levels and sublevels in arrangements -- and, dually, $k$-sets and $(\leq k)$-sets -- are fundamental notions in discrete and computational geometry and natural generalizations of convex polytopes, which correspond to the $0$-level. A long-standing conjecture of Eckhoff, Linhart, and Welzl, which would generalize McMullen's Upper Bound Theorem for polytopes and provide an exact refinement of asymptotic bounds by Clarkson, asserts that for all $k\leq \lfloor \frac{n-d-2}{2}\rfloor$, the number of $(\leq k)$-sets of a set $S$ of $n$ points in $\mathbf{R}^d$ is maximized if $S$ is the vertex set of a neighborly polytope. As a new tool for studying this conjecture and related problems, we introduce the $g$-matrix, which generalizes both the $g$-vector of a simple polytope and a Gale dual version of the $g$-vector studied by Lee and Welzl. Our main result is that the $g$-matrix of every vector configuration in $\mathbf{R}^3$ is non-negative, which implies the Eckhoff--Linhart--Welzl conjecture in the case where $d=n-4$. As a corollary, we obtain the following result about crossing numbers: Consider a configuration $V\subset S^2 \subset \mathbf{R}^3$ of $n$ unit vectors, and connect every pair of vectors by the unique shortest geodesic arc between them in the unit sphere $S^2$. This yields a drawing of the complete graph $K_n$ in $S^2$, which we call a spherical arc drawing. Complementing previous results for rectilinear drawings, we show that the number of crossings in any spherical arc drawing of $K_n$ is at least $\frac{1}{4}\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor \lfloor \frac{n-2}{2}\rfloor \lfloor \frac{n-3}{2}\rfloor$, which equals the conjectured value of the crossing number of $K_n$. Moreover, the lower bound is attained if $V$ is coneighborly, i.e., if every open linear halfspace contains at least $\lfloor (n-2)/2 \rfloor$ of the vectors in $V$.

Sublevels in arrangements and the spherical arc crossing number of complete graphs

TL;DR

The paper develops a unified framework for sublevels in 3D arrangements via the g-matrix, establishing nonnegativity results that imply the Eckhoff–Linhart–Welzl conjecture for . It then connects these combinatorial bounds to spherical arc drawings of , proving a universal lower bound on the crossing number that matches Hill's conjectured value for coneighborly configurations. Gale duality ties -bounds to sublevel bounds, yielding exact upper bounds in rank and clarifying when equality is achieved. Overall, the work links generalized upper bound phenomena, mutation-driven changes, and geometric dualities to derive precise combinatorial constraints and optimal configuration structures.

Abstract

Levels and sublevels in arrangements -- and, dually, -sets and -sets -- are fundamental notions in discrete and computational geometry and natural generalizations of convex polytopes, which correspond to the -level. A long-standing conjecture of Eckhoff, Linhart, and Welzl, which would generalize McMullen's Upper Bound Theorem for polytopes and provide an exact refinement of asymptotic bounds by Clarkson, asserts that for all , the number of -sets of a set of points in is maximized if is the vertex set of a neighborly polytope. As a new tool for studying this conjecture and related problems, we introduce the -matrix, which generalizes both the -vector of a simple polytope and a Gale dual version of the -vector studied by Lee and Welzl. Our main result is that the -matrix of every vector configuration in is non-negative, which implies the Eckhoff--Linhart--Welzl conjecture in the case where . As a corollary, we obtain the following result about crossing numbers: Consider a configuration of unit vectors, and connect every pair of vectors by the unique shortest geodesic arc between them in the unit sphere . This yields a drawing of the complete graph in , which we call a spherical arc drawing. Complementing previous results for rectilinear drawings, we show that the number of crossings in any spherical arc drawing of is at least , which equals the conjectured value of the crossing number of . Moreover, the lower bound is attained if is coneighborly, i.e., if every open linear halfspace contains at least of the vectors in .

Paper Structure

This paper contains 9 sections, 17 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Dependency and Dissection Patterns
  3. The Spherical Arc Crossing Number of $K_n$
  4. The $g$-Matrix
  5. Bounding the Spherical Arc Crossing Number
  6. Gale Duality and Exact Upper Bounds for Sublevels
  7. Continuous Motion and the $g$-Matrix
  8. The Upper Bound for $g_{1,k}^*$ in Rank $3$
  9. The Remaining Calculation for the Spherical Arc Crossing Number

Key Result

Theorem 2

Let $V = \{ v_1, \ldots, v_n \} \subset \mathbb{R}^3$ be a vector configuration in general position. Then, for all $s\leqslant n$, the numbers $f^*_{s,0}(V)$ and $f^*_{s,\leqslant 1}(V):=f^*_{s,0}(V)+f^*_{s,1}(V)$ are maximized if $V$ is coneighborly.

Figures (2)

  • Figure 1: A mutation between an antipodal pair of triangles $\sigma,-\sigma$ of types $(3,k)$ and $(0,n-3-k)$ (left column top and bottom) and an antipodal pair $\tau,-\tau$ of triangles of types $(0,k)$ and $(3,n-3-k)$ (right column top and bottom). The horizontal arrows are marked with the types of the appearing triangles as we move between the left and the right column. The labels in the full-dimensional cells indicate their levels, and the little arrows indicate positive hemispheres.
  • Figure 2: A mutation between an antipodal pair of triangles $\sigma,-\sigma$ of types $(2,k)$ and $(1,n-3-k)$ (left column) and an antipodal pair $\tau,-\tau$ of triangles of types $(1,k)$ and $(2,n-3-k)$ (right column).

Theorems & Definitions (30)

  • Definition 1: $f$ and $f^*$
  • Theorem 2
  • Conjecture 3: Hill
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Remark 7
  • Example 8: Cyclic and Cocyclic Configurations
  • Definition 9: $g$-matrix and $g^*$-matrix
  • Lemma 10
  • ...and 20 more