Barycentric Cuts Through a Convex Body

Let K be a convex body in \(\mathbb {R}^n\) (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 \(K \cap h\). 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 \(p=p_0\) 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 \(n \ge 2\), there are always at least three distinct barycentric cuts through the point \(p_0 \in K\) of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through \(p_0\) are guaranteed if \(n \ge 3\).

1. Given \(v, v' \in S^{n-1}\), \(\lambda (H_v \cap K)\) and \(\lambda (H_{v'} \cap K)\) differ by at most \(\lambda ((H_v \Delta H_{v'})\cap K)\) where \(\Delta \) is the symmetric difference. For \(\varepsilon > 0\) and v and \(v'\) sufficiently close, \(\lambda ((H_v \Delta H_{v'}) \cap K) < \varepsilon \lambda (K)\) as K is bounded.

2. We remark that our depth function slightly differs from the function f(H, p) used by Grünbaum [12, § 6.2]. However, the point of maximal depth coincides with the ‘critical point’ in [12] and hyperplanes realizing the depth for \(p_0\) coincide with the ‘hyperplanes through the critical point dividing the volume of K in the ratio \(F_2(K)\)’.

3. The idea of the proof is simple: For contradiction assume that h realizes the depth of p but that the barycenter b of \(K \cap h\) differs from p. Let \(v \in S^{n-1}\) be such that \(h = h_v\) and \({{\,\mathrm{depth}\,}}(p, K) = \delta ^p(v)\). Consider the affine \((d-2)\)-space \(\rho \) in h passing through p and perpendicular to the segment bp. Then by a small rotation of h along \(\rho \) we can get \(h_{v'}\) such that \(\delta ^p(v') < \delta ^p(v)\) which contradicts that h realizes the depth of p. Of course, it remains to check the details.

4. We remark that the second condition in the statement of the result in [19] is equivalent to the statement that \(0 \in {{\,\mathrm{conv}\,}}U\), in our notation.

5. Sketch of the inverse ray basis theorem: if there is a closed hemisphere \(C \subseteq S^{n-1}\) which does not contain a point of U, let v be the center of C. Then a small shift of \(p_0\) in the direction of v yields a point of larger depth, a contradiction.

6. When compared with formula (3.1) in [13], we obtain a different sign in front of the integral. This is caused by integration over the opposite halfspace.

7. By a metric ball we mean a ball with a given center and radius. This way, we distinguish a metric ball from a general topological ball.

8. We point out that the current online version of [14] contains a different proof of Proposition 1.14. Therefore, here we refer to the printed version of the book.

We thank Stanislav Nagy for introducing us to Grünbaum’s questions, for useful discussions on the topic, for providing us with many references, and for comments on a preliminary version of this paper. We thank Jan Kynčl and Pavel Valtr for letting us know about a more general counterexample they found. We thank Roman Karasev for providing us with references [1, 15] and for comments on a preliminary version of this paper. Finally, we thank an anonymous referee for many comments on a preliminary version of the paper which, in particular, yielded an important correction in Sect. 4.

Authors and Affiliations

1. Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

   Zuzana Patáková

2. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

   Martin Tancer

3. IST Austria, Klosterneuburg, Austria

   Uli Wagner

Corresponding author

Correspondence to Martin Tancer.

Editor in Charge: Kenneth Clarkson

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work by Zuzana Patáková has been partially supported by Charles University Research Center Program No. UNCE/SCI/022, and part of it was done during her research stay at IST Austria. The work by Martin Tancer is supported by the GAČR Grant 19-04113Y and by the Charles University Projects PRIMUS/17/SCI/3 and UNCE/SCI/004.

Reprints and permissions