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=po 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 po ELEMENT OF K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through po are guaranteed if n >= 3.