Linear relations between face numbers of levels in arrangements

TL;DR

This work studies linear relations among face numbers of levels in spherical hemispherical arrangements, encoding data in the f-matrix f(V) and the f-polynomial f_V(x,y). Central to the approach is the g-matrix g(V\to W), which records how f changes under mutations along continuous deformations between vector configurations; the f-difference is recovered via a linear transformation from g, and g is injectively determined by this difference. The authors determine the affine space ound_{n,r} spanned by all f-matrices for configurations of n vectors in general position in r^r, and the subspace ound^0_{n,r} for pointed configurations, with precise dimension formulas; they also describe Gale-dual results for f^* and Radon partitions. The framework unifies Dehn-Sommerville-type relations for levels, neighborly and coneighborly configurations, and contractions/deletions, yielding structural descriptions of all first-principle linear relations among level face numbers and their Radon-partition analogues, with implications for oriented matroids and combinatorial geometry.

Abstract

We study linear relations between face numbers of levels in arrangements. Let $V = \{ v_1, \ldots, v_n \} \subset \mathbf{R}^{r}$ be a vector configuration in general position, and let $\mathcal{A}(V)$ be polar dual arrangement of hemispheres in the $d$-dimensional unit sphere $S^d$, where $d=r-1$. For $0\leq s \leq d$ and $0 \leq t \leq n$, let $f_{s,t}(V)$ denote the number of faces of \emph{level} $t$ and dimension $d-s$ in the arrangement $\mathcal{A}(V)$ (these correspond to partitions $V=V_-\sqcup V_0 \sqcup V_+$ by linear hyperplanes with $|V_0|=s$ and $|V_-|=t$). We call the matrix $f(V):=[f_{s,t}(V)]$ the \emph{$f$-matrix} of $V$. Completing a long line of research on linear relations between face numbers of levels in arrangements, we determine, for every $n\geq r \geq 1$, the affine space $\mathfrak{F}_{n,r}$ spanned by the $f$-matrices of configurations of $n$ vectors in general position in $\mathbf{R}^r$; moreover, we determine the subspace $\mathfrak{F}^0_{n,r} \subset \mathfrak{F}_{n,r}$ spanned by all \emph{pointed} vector configurations (i.e., such that $V$ is contained in some open linear halfspace), which correspond to point sets in $\mathbf{R}^d$. This generalizes the classical fact that the Dehn--Sommerville relations generate all linear relations between the face numbers of simple polytopes (the faces at level $0$) and answers a question posed by Andrzejak and Welzl in 2003. The key notion for the statements and the proofs of our results is the $g$-matrix of a vector configuration, which determines the $f$-matrix and generalizes the classical $g$-vector of a polytope. By Gale duality, we also obtain analogous results for partitions of vector configurations by sign patterns of nontrivial linear dependencies, and for \emph{Radon partitions} of point sets in $\mathbf{R}^d$.

Figures (2)

• Figure 1: A mutation of Type $(0, k) \equiv (3, n-3-k)$ (from left to right), respectively $(3, k)\equiv(0, n-3-k)$ (from right to left) in $S^2$. The upper row shows the triangular faces $\sigma$ and $\tau$ before and after the mutation, and the lower row shows the corresponding antipodal faces $-\sigma$ and $-\tau$. The little arrows indicate positive halfspaces, and the labels in full-dimensional faces indicate levels.
• Figure 2: A mutation of Type $(1, k)\equiv (2, n-3-k)$ (from left to right), respectively $(2, k)\equiv(1, n-3-k)$ (from right to left) in $S^2$.

Theorems & Definitions (33)

• Definition 1: $f$-matrix and $f$-polynomial
• Lemma 3
• Theorem 4: Dehn--Sommerville Relations for Levels in Simple Arrangements
• Remark 5
• Theorem 6
• Remark 7
• Remark 8
• Theorem 9
• Theorem 10
• Remark 11