Stabbing simplices by points and flats

The following result was proved by Bárány in 1982: For every d≥1, there exists c_d>0 such that for every n-point set S in ℝ^d, there is a point p∈ℝ^d contained in at least c_dn^d+1-O(n^d) of the d-dimensional simplices spanned by S. We investigate the largest possible value of c_d. It was known that c_d≤1/(2^d(d+1)!) (this estimate actually holds for every point set S). We construct sets showing that c_d≤(d+1)^-(d+1), and we conjecture that this estimate is tight. The best known lower bound, due to Wagner, is c_d≥γ_d:=(d²+1)/((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 γ_dn^d+1+O(n^d) 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^d, there exists a (d-2)-flat that stabs at least c_d,d-2n³-O(n²) of the triangles spanned by S, with c_d,d-2≥1/24(1-1/(2d-1)²). 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^d can be divided into 4d-2 equal parts by 2d-1 hyperplanes intersecting in a common (d-2)-flat.