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 $d\leq 4$.