Sublevels in arrangements and the spherical arc crossing number of complete graphs
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2025
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| _version_ | 1866916683758698496 |
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| author | Streltsova, Elizaveta Wagner, Uli |
| author_facet | Streltsova, Elizaveta Wagner, Uli |
| contents | 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 $V$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_07770 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sublevels in arrangements and the spherical arc crossing number of complete graphs Streltsova, Elizaveta Wagner, Uli Combinatorics Computational Geometry 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 $V$. |
| title | Sublevels in arrangements and the spherical arc crossing number of complete graphs |
| topic | Combinatorics Computational Geometry |
| url | https://arxiv.org/abs/2504.07770 |