Stabbing Simplices by Points and Flats

The following result was proved by Barany 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 is an element of R-d contained in at least cdn(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) >= gamma d := (d(2) + 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 gamma 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 subset of R-d, there exists a (d - 2)-flat that stabs at least c(d,d-2)n(3) - O(n(2)) 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.