The Minimum Number of Peeling Sequences of a Point Set.
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| Title: | The Minimum Number of Peeling Sequences of a Point Set. |
|---|---|
| Authors: | Simon, Dániel G.1,2 (AUTHOR) dgsimon@renyi.hu |
| Source: | Discrete & Computational Geometry. Jun2026, Vol. 75 Issue 4, p1122-1133. 12p. |
| Subjects: | Computational geometry, Combinatorial geometry, Mathematics, Point set theory, Mathematical bounds |
| Abstract: | Let P be a set of n points in R d , in general position. We remove all of them one by one, in each step erasing one vertex of the convex hull of the current remaining set. Let g d (P) denote the number of different removal orders we can attain while erasing all points of P this way, and let g d (n) be the minimum of g d (P) over all n-element point sets P ⊂ R d . Dumitrescu and Tóth showed that g d (n) ≤ (d + 1) (d + 1) 2 n . We substantially improve their bound, by proving that g d (n) = O ((d + d ln d) (2 + (d - 1) ⌊ d ln d ⌋) n) . It follows that, for any ϵ > 0 , there exist sufficiently high dimensional point sets P ⊂ R d with g d (P) ≤ O (d (2 + ϵ) n) . This almost closes the gap between the upper bound and the best-known lower bound (d + 1) n for large values of d. [ABSTRACT FROM AUTHOR] |
| Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 194162858 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: The Minimum Number of Peeling Sequences of a Point Set. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Simon%2C+Dániel+G%2E%22">Simon, Dániel G.</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> dgsimon@renyi.hu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Jun2026, Vol. 75 Issue 4, p1122-1133. 12p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Computational+geometry%22">Computational geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Combinatorial+geometry%22">Combinatorial geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics%22">Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Point+set+theory%22">Point set theory</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+bounds%22">Mathematical bounds</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Let P be a set of n points in R d , in general position. We remove all of them one by one, in each step erasing one vertex of the convex hull of the current remaining set. Let g d (P) denote the number of different removal orders we can attain while erasing all points of P this way, and let g d (n) be the minimum of g d (P) over all n-element point sets P ⊂ R d . Dumitrescu and Tóth showed that g d (n) ≤ (d + 1) (d + 1) 2 n . We substantially improve their bound, by proving that g d (n) = O ((d + d ln d) (2 + (d - 1) ⌊ d ln d ⌋) n) . It follows that, for any ϵ > 0 , there exist sufficiently high dimensional point sets P ⊂ R d with g d (P) ≤ O (d (2 + ϵ) n) . This almost closes the gap between the upper bound and the best-known lower bound (d + 1) n for large values of d. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s00454-024-00713-2 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 12 StartPage: 1122 Subjects: – SubjectFull: Computational geometry Type: general – SubjectFull: Combinatorial geometry Type: general – SubjectFull: Mathematics Type: general – SubjectFull: Point set theory Type: general – SubjectFull: Mathematical bounds Type: general Titles: – TitleFull: The Minimum Number of Peeling Sequences of a Point Set. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Simon, Dániel G. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 01795376 Numbering: – Type: volume Value: 75 – Type: issue Value: 4 Titles: – TitleFull: Discrete & Computational Geometry Type: main |
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