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.
Full text of the article: