Abstract
Let K be a convex body in Rn (i.e., a compact convex set with nonempty interior).
Given a point p in the interior of K, a hyperplane h passing through p is called barycentric if p is the barycenter of K INTERSECTION h. In 1961, Grünbaum raised the question whether, for every K, there exists an interior point p through which there are at least n + 1 distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if p = p0 is the point of maximal depth in K. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum's question. It follows from known results that for n >= 2, there are always at least three distinct barycentric cuts through the point p0 ELEMENT OF K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through p0 are guaranteed if n >= 3.