research-article

Convex Hulls of Random Order Types

Authors:

Emo WelzlAuthors Info & Claims

Journal of the ACM, Volume 70, Issue 1

Article No.: 8, Pages 1 - 47

https://doi.org/10.1145/3570636

Published: 16 January 2023 Publication History

Abstract

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3):

(a)

The number of extreme points in an n-point order type, chosen uniformly at random from all such order types, is on average 4+o(1). For labeled order types, this number has average \(4- \mbox{$\frac{8}{n^2 - n +2}$}\) and variance at most 3.

(b)

The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size.

Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2d+o (1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods show the following relative of the Erdős-Szekeres theorem: for any fixed k, as n → ∞, a proportion 1 - O(1/n) of the n-point simple order types contain a triangle enclosing a convex k-chain over an edge.

For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of A₄, S₄, or A₅ (and each case is possible). These are the finite subgroups of SO(3) and our proof follows the lines of their characterization by Felix Klein.

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Index Terms

Convex Hulls of Random Order Types
1. Theory of computation
  1. Randomness, geometry and discrete structures

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cover image Journal of the ACM

Journal of the ACM Volume 70, Issue 1

February 2023

405 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3572730

Editor:
Venkatesan Guruswami
University of California, Berkeley, United States

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Publication History

Published: 16 January 2023

Online AM: 02 December 2022

Accepted: 07 October 2022

Revised: 31 May 2022

Received: 15 September 2020

Published in JACM Volume 70, Issue 1

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Cited By

Steiner R(2024)A Logarithmic Bound for Simultaneous Embeddings of Planar GraphsDiscrete & Computational Geometry10.1007/s00454-024-00665-7Online publication date: 4-Jun-2024
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Dobbins MLee S(2024)Inscribable Order TypesDiscrete & Computational Geometry10.1007/s00454-023-00591-072:2(704-727)Online publication date: 1-Sep-2024
https://dl.acm.org/doi/10.1007/s00454-023-00591-0
Steiner R(2024)A Logarithmic Bound for Simultaneous Embeddings of Planar GraphsGraph Drawing and Network Visualization10.1007/978-3-031-49275-4_9(133-140)Online publication date: 6-Jan-2024
https://doi.org/10.1007/978-3-031-49275-4_9
Bárány IPór A(2023)Orientation Preserving Maps of the Square Grid IIDiscrete & Computational Geometry10.1007/s00454-023-00531-y72:2(569-602)Online publication date: 5-Oct-2023
https://doi.org/10.1007/s00454-023-00531-y

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