Improved Bounds for Point Selections and Halving Hyperplanes in Higher Dimensions

Let (P, E) be a (d + 1)-uniform geometric hypergraph, where P is an n-point set in general position in R^d and E ⊆ ^!_d^P₊₁^" is a collection of !^!_d+1^n" d-dimensional simplices with vertices in P, for 0 < ! ≤ 1. We show that there is a point x ∈ R^d that pierces (equation presented) simplices in E, for any fixed δ > 0. This is a dramatic improvement in all dimensions d ≥ 3, over the previous lower bounds of the general form (equation presented), which date back to the seminal 1991 work of Alon, Bárány, Füredi and Kleitman. As a result, any n-point set in general position in R^d admits only (equation presented) halving hyperplanes, for any δ > 0, which is a significant improvement over the previously best known bound (equation presented) in all dimensions d ≥ 5. An essential ingredient of our proof is the following semi-algebraic Turán-type result of independent interest, which holds for any fixed δ > 0: Let (V1, ..., V_k, E) be a hypergraph of bounded semi-algebraic description complexity, whose vertices are in sufficiently general position in R^d. Suppose that |E| ≥ ε|V1|·...·|V_k| holds for some ε > 0, then there exist subsets (equation presented).