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International Computing and Combinatorics Conference

COCOON 2001: Computing and Combinatorics pp 111–120Cite as

An Improved Upper Bound on the Size of Planar Convex-Hulls

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2108))

Abstract

Let C be the convex-hull of a set of points S with integral coordinates in the plane. It is well-known that |C| ≤ cD 2/3 for some constant c where D is the diameter of S: i.e. the maximum distance between any pair of points in S. It has been shown that c = 7.559.. for an arbitrary S, and c = 3.496.. in the special case when S is a ball centered at the origin in the plane. In this paper we show that c = 12/ 3v 4p2 = 3.524.. is sufficient for an arbitrary set of lattice points S of diameter D in the plane, and |C| ~ 12 3v2/(9p2) D 2/3 = 3.388..D 2/3 is achieved asymptotically. Our proof is based on the construction of a special set in ?rst quadrant, and the analysis of the result involves the calculation of the average order of certain number-theoretical functions associated with the Euler totient function φ(n).

Supported in part by NSF Grant No. CCR-9821038, and a UCSB-COR faculty research grant.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of California, Santa Barbara, CA, 93106, USA

    Abdullah N. Arslan & Ömer Eğecioğlu

Authors
  1. Abdullah N. Arslan
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  2. Ömer Eğecioğlu
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Massachusetts, One University Avenue, Lowell, MA, 01854, USA

    Jie Wang

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© 2001 Springer-Verlag Berlin Heidelberg

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Arslan, A.N., Eğecioğlu, Ö. (2001). An Improved Upper Bound on the Size of Planar Convex-Hulls. In: Wang, J. (eds) Computing and Combinatorics. COCOON 2001. Lecture Notes in Computer Science, vol 2108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44679-6_13

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  • DOI: https://doi.org/10.1007/3-540-44679-6_13

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42494-9

  • Online ISBN: 978-3-540-44679-8

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