Stabbing simplices by points and flats

The following result was proved by Bárány in 1982: For every d ≥ 1 there exists cd > 0 such that for every n-point set S in R d there is a point p ∈ R contained in at least cdn d+1 −O(n) of the d-dimensional simplices spanned by S. We investigate the largest possible value of cd. It was known that cd ≤ 1/(2(d+ 1)!) (this estimate actually holds for every point set S). We construct sets showing that cd ≤ (d + 1), and we conjecture this estimate to be tight. The best known lower bound, due to Wagner, is cd ≥ γd := (d + 1)/((d + 1)!(d + 1)); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than γdn d+1 +O(n) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved. We also prove that for every n-point set S ⊂ R there exists a (d − 2)-flat that stabs at least cd,d−2n 3 − O(n) of the triangles spanned by S, with cd,d−2 ≥ 1 24 (1− 1/(2d− 1) ). To this end, we establish an equipartition result of independent interest (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution in R can be divided into 4d − 2 equal parts by 2d − 1 hyperplanes intersecting in a common (d− 2)-flat.