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PreprintCC BY 4.0arXiv

A centerpoint theorem for three planar convex bodies

  • Hongyu Cheng
  • Amitabh Basu
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Abstract

For planar convex bodies A0,A1,A2 satisfying 1/2(A0+A2)⊆ A1, we prove that the union of the three slices {i}× Ai⊆ℝ3, 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×ℝ2 with the best possible constant.

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Cite

BibTeXRIS
BibTeX
@misc{cheng2026centerpoint,
  title = {{A centerpoint theorem for three planar convex bodies}},
  author = {Hongyu Cheng and Amitabh Basu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32953},
  eprint = {2609.32953},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32953},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A centerpoint theorem for three planar convex bodies
AU  - Hongyu Cheng
AU  - Amitabh Basu
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32953
UR  - https://arxiv.org/abs/2609.32953
AB  - For planar convex bodies A_0,A_1,A_2 satisfying 1/2(A_0+A_2)⊆ A_1, we prove that the union of the three slices {i}× A_i⊆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×ℝ^2 with the best possible constant.
ER  -