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 ℝd, there is a point p∈ℝd contained in at least cdnd+1−O(nd) of the d-dimensional simplices spanned by S.
We investigate the largest possible value of cd. It was known that cd≤1/(2d(d+1)!) (this estimate actually holds for every point set S). We construct sets showing that cd≤(d+1)−(d+1), and we conjecture that this estimate is tight. The best known lower bound, due to Wagner, is cd≥γd:=(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 γdnd+1+O(nd) 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⊂ℝd, there exists a (d−2)-flat that stabs at least cd,d−2n3−O(n2) of the triangles spanned by S, with $c_{d,d-2}\ge\frac{1}{24}(1-1/(2d-1)^{2})$ . 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 ℝd can be divided into 4d−2 equal parts by 2d−1 hyperplanes intersecting in a common (d−2)-flat.