New lower bounds for halfspace emptiness

The author derives a lower bound of Ω(n^4/3) for the halfspace emptiness problem: given a set of n points and n hyperplanes in R⁵, is every point above every hyperplane? This matches the best known upper bound to within polylogarithmic factors, and improves the previous best lower bound of Ω(nlogn). The lower bound applies to partitioning algorithms in which every query region is a polyhedron with a constant number of facets