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arXiv · 2609.32953

A centerpoint theorem for three planar convex bodies

Hongyu Cheng·Amitabh Basu
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Abstract

For planar convex bodies $A_0,A_1,A_2$ satisfying $\frac12(A_0+A_2)\subseteq A_1$, we prove that the union of the three slices $\{i\}\times A_i\subseteq\mathbb{R}^3$, $i=0,1,2$, 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 $\mathbb{Z}\times\mathbb{R}^2$ with the best possible constant.

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Hongyu Cheng, Amitabh Basu. 2026-09-26. A centerpoint theorem for three planar convex bodies. https://arxiv.org/abs/2609.32953

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