Discrepancy theory, Tverberg's theorem, and regression depth

@misc{lopez2026discrepancy,
  title = {{Discrepancy theory, Tverberg's theorem, and regression depth}},
  author = {Aleksey Lopez and Pablo Soberón},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24775},
  eprint = {2609.24775},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24775},
}

TY  - UNPB
TI  - Discrepancy theory, Tverberg's theorem, and regression depth
AU  - Aleksey Lopez
AU  - Pablo Soberón
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24775
UR  - https://arxiv.org/abs/2609.24775
AB  - We prove new bounds for Tverberg's theorem with tolerance. We show that N = rt+Θ_d,r(t^1/2−1/(2d)), where N is the smallest number such that any set of N points in ℝ^d has a partition into r parts such that the convex hulls of the parts intersect even if we remove any t of the points. We extend Tverberg's theorem with tolerance to families of hyperplanes in ℝ^d, and show that for any set of rt + O_d,r(t^1/2−1/(2d)√(log (t+1))) hyperplanes in ℝ^d there exists a partition of them into r parts such that the regression hulls of the parts intersect even if any t hyperplanes are removed. Our bounds follow from establishing a connection between Tverberg-type results and discrepancy theory.
ER  -