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PreprintCC BY-SA 4.0arXiv

Peeling sequences: a directional method for the three-block construction

  • Dániel Gábor Simon
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Abstract

A peeling sequence of a finite planar point set is an ordering of its points, such that when the points get removed in this order, each removed point is a vertex of the current convex hull of the set. Write g(S) for the number of such sequences, and g(n) for the minimum of g(S) over n-point sets in general position. We present a method which can be used to prove better upper bounds on the previously analysed recursive 3-block constructions Sn. In fact, we prove g(n)≤ g(Sn)=O(6.57n), using directional restrictions and a weighted prefix-tree form of the Kraft--McMillan inequality. From the other direction, we show that the same construction cannot attain the current best known lower bound of c·3n. In fact, g(Sn)≥ 5.22n when n is a large enough power of $3$. Improving the general lower bound on g(n) is still an open question.

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Cite

BibTeXRIS
BibTeX
@misc{simon2026peeling,
  title = {{Peeling sequences: a directional method for the three-block construction}},
  author = {Dániel Gábor Simon},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.19540},
  eprint = {2609.19540},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19540},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Peeling sequences: a directional method for the three-block construction
AU  - Dániel Gábor Simon
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.19540
UR  - https://arxiv.org/abs/2609.19540
AB  - A peeling sequence of a finite planar point set is an ordering of its points, such that when the points get removed in this order, each removed point is a vertex of the current convex hull of the set. Write g(S) for the number of such sequences, and g(n) for the minimum of g(S) over n-point sets in general position. We present a method which can be used to prove better upper bounds on the previously analysed recursive 3-block constructions S_n. In fact, we prove g(n)≤ g(S_n)=O(6.57^n), using directional restrictions and a weighted prefix-tree form of the Kraft--McMillan inequality. From the other direction, we show that the same construction cannot attain the current best known lower bound of c·3^n. In fact, g(S_n)≥ 5.22^n when n is a large enough power of $3$. Improving the general lower bound on g(n) is still an open question.
ER  -