Barycentric Cuts at the Point of Maximal Depth

Xiaoxiang Jiao; Hangyue Zhu

Barycentric Cuts at the Point of Maximal Depth

Metric Geometry 2026-08-03 v1

Authors: Xiaoxiang Jiao , Hangyue Zhu

Abstract

We determine the sharp number of barycentric hyperplanes through the point of maximal halfspace depth of a convex body. More precisely, we prove that $\nu(n)=4\qquad\text{for every $n\ge 3$.}$ This completely resolves the maximal-depth version of the barycentric section problem.

Keywords

barycentric hyperplanes halfspace depth convex body barycentric section problem maximal depth

Cite

@article{arxiv.2608.01721,
  title  = {Barycentric Cuts at the Point of Maximal Depth},
  author = {Xiaoxiang Jiao and Hangyue Zhu},
  journal= {arXiv preprint arXiv:2608.01721},
  year   = {2026}
}