A centerpoint theorem for three planar convex bodies

Hongyu Cheng; Amitabh Basu

paper

A centerpoint theorem for three planar convex bodies

arXiv:2609.32953

Abstract

For planar convex bodies $A_{0}, A_{1}, A_{2}$ satisfying $\frac{1}{2} (A_{0} + A_{2}) \subseteq A_{1}$ , we prove that the union of the three slices ${i} \times A_{i} \subseteq R^{3}$ , $i = 0, 1, 2$ , contains a point such that every closed halfspace containing it captures at least $2/9$ of their total area. This establishes the three-slice case of Oertel's mixed-integer centerpoint conjecture in $Z \times R^{2}$ with the best possible constant.