Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 35 (1994), No. 1, 29-35.
The Upper Bound Conjecture for Arrangements of Halfspaces
Johann Linhart
Abstract.
For an arbitrary arrangement of $n$ open hemispheres of $S^d$, it is
conjectured that the number of vertices contained in at most $k$ of the
hemispheres attains its maximum for each $k<(n-d)/2$ in case the hemispheres
determine the dual of a spherical cyclic polytope. This would imply a sharp
upper bound on the
analogous numbers for arrangements of half-spaces in $E^d$.
The latter is proved here for $d\leq 4$.