Barycentric Cuts at Maximal Depth and a Five-Cut Theorem

Abstract

We determine the sharp number and the complete finite spectrum of barycentric hyperplanes through the Tukey median. We further prove that every convex body in ${\mathbb{R}}^n$, $n\ge 3$, contains an interior point incident with at least five barycentric hyperplanes. In particular, this settles Grünbaum's original conjecture in dimension four.