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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 2, pp. 431-437 (2001)
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The Upper Bound Conjecture for Arrangements of Halfspaces

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Gerhard Wesp

Institut für Mathematik, Universität Salzburg, Hellbrunnerstr. 34, A-5020 Salzburg, e-mail: gwesp@cosy.sbg.ac.at

**Abstract:** Let $\cal A$ be an arrangement of $n$ open halfspaces in $** R**^{r-1}$. In [L], Linhart proved that for $r\le5$, the numbers of vertices of $\cal A$ contained in at most $k$ halfspaces are bounded from above by the corresponding numbers of ${\cal C}(n,r)$, where ${\cal C}(n,r)$ is an arrangement realizing the alternating oriented matroid of rank $r$ on $n$ elements. In the present paper Linhart's result is generalized to faces of dimension $s-1$ for $1\le s\le 4$. \item{[L]} Linhart, Johann: * The Upper Bound Conjecture for arrangements of halfspaces.* Beiträge Algebra Geom. ** 35**(1) (1994), 29-35.

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