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Line arrangements, bounded cells, and discrepancy

  • Imre Bárány
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Abstract

An arrangement of n lines in the plane defines cells, i.e., connected components of the complement of the arrangement. Answering a question of Volkmar Welker we show that, apart from two special cases, there is always a line L in the arrangement such that one of the open halfplanes whose boundary is L contains more than half of the bounded cells of the arrangement. We also consider some unusual discrepancy problems that are related to this question.

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Cite

BibTeXRIS
BibTeX
@misc{barany2026line,
  title = {{Line arrangements, bounded cells, and discrepancy}},
  author = {Imre Bárány},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23203},
  eprint = {2609.23203},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23203},
}
RIS
TY  - UNPB
TI  - Line arrangements, bounded cells, and discrepancy
AU  - Imre Bárány
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.23203
UR  - https://arxiv.org/abs/2609.23203
AB  - An arrangement of n lines in the plane defines cells, i.e., connected components of the complement of the arrangement. Answering a question of Volkmar Welker we show that, apart from two special cases, there is always a line L in the arrangement such that one of the open halfplanes whose boundary is L contains more than half of the bounded cells of the arrangement. We also consider some unusual discrepancy problems that are related to this question.
ER  -