🤖 AI Summary
This work characterizes the affine space $mathcal{F}_{n,r}$ spanned by the $f$-matrices (matrices of face numbers across dimensions) of all configurations of $n$ $r$-dimensional vectors in general position, and its subspace $mathcal{F}^0_{n,r}$ corresponding to pointed subconfigurations (i.e., those lying in some open linear halfspace). Using Gale duality, polar dual arrangements, sign pattern analysis, and Radon partition theory, the dimension and combinatorial structure of $mathcal{F}_{n,r}$ are fully determined for the first time. A generalized $g$-matrix framework is introduced to unify the description of face-number dependencies, and the Dehn–Sommerville relations are systematically extended to arbitrary levels. Finally, a complete system of linear relations among spherical arrangement face numbers—across all dimensions—is established for point sets, thereby resolving the open problem posed by Andrzejak and Welzl (2003).
📝 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 $0leq s leq d$ 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 $ngeq r geq 1$, 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 $mathbf{R}^d$.