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The upper envelope of piecewise linear functions: Tight bounds on the number of faces

Discrete & Computational Geometry volume 4, pages 337–343 (1989)Cite this article

Abstract

This note proves that the maximum number of faces (of any dimension) of the upper envelope of a set ofn possibly intersectingd-simplices ind+1 dimensions is Θ(n dα(n)). This is an extension of a result of Pach and Sharir [PS] who prove the same bound for the number ofd-dimensional faces of the upper envelope.

References

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Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Urbana-Champaign, 61801, Urbana, IL, USA

    Herbert Edelsbrunner

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  1. Herbert Edelsbrunner
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Additional information

This work was supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by the National Science Foundation under Grant CCR-8714565. Research on the presented result was partially carried out while the author worked for the IBM T. J. Watson Research Center at Yorktown Height, New York, USA.

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Edelsbrunner, H. The upper envelope of piecewise linear functions: Tight bounds on the number of faces. Discrete Comput Geom 4, 337–343 (1989). https://doi.org/10.1007/BF02187734

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  • DOI: https://doi.org/10.1007/BF02187734

Keywords

  • Recurrence Relation
  • Euler Characteristic
  • Discrete Comput Geom
  • Cell Complex
  • Piecewise Linear Function