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. |
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| 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] |
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| Database: | Engineering Source |
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| 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] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-024-00713-2 |
