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Characterization of Extremal Antipodal Polygons

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Abstract

Let \(S\) be a set of \(2n\) points on a circle such that for each point \(p \in S\) also its antipodal (mirrored with respect to the circle center) point \(p'\) belongs to \(S\). A polygon \(P\) of size \(n\) is called antipodal if it consists of precisely one point of each antipodal pair \((p,p')\) of \(S\). We provide a complete characterization of antipodal polygons which maximize (minimize, respectively) the area among all antipodal polygons of \(S\). Based on this characterization, a simple linear time algorithm is presented for computing extremal antipodal polygons. Moreover, for the generalization of antipodal polygons to higher dimensions we show that a similar characterization does not exist.

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Notes

  1. This property is not “if and only if” because there also exist non-thick polygons fulfilling the property.

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Acknowledgments

The problems studied here were introduced and partially solved during a visit to the University of La Havana, Cuba. We thank the project COFLA: Computational analysis of the Flamenco music (FEDER P09-TIC-4840 and FEDER P12-TIC-1362) for posing us the basic problem studied in this paper.

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Authors and Affiliations

  1. Institute for Software Technology, University of Technology Graz, Graz, Austria

    O. Aichholzer

  2. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    L. E. Caraballo & J. M. Díaz-Báñez

  3. Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Mexico City, Mexico

    R. Fabila-Monroy

  4. Departamento de Ciencias de la Computación (DCC), Universidad de Chile, Santiago, Chile

    C. Ochoa

  5. University of Technology Graz, Graz, Austria

    P. Nigsch

Authors
  1. O. Aichholzer
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  2. L. E. Caraballo
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  3. J. M. Díaz-Báñez
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  4. R. Fabila-Monroy
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  5. C. Ochoa
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  6. P. Nigsch
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Corresponding author

Correspondence to J. M. Díaz-Báñez.

Additional information

O. Aichholzer was partially supported by the ESF EUROCORES programme EuroGIGA - ComPoSe, Austrian Science Fund (FWF): I 648-N18. J.M.D.-B. was partially supported by projects FEDER P09-TIC-4840, P12-TIC-1362 (Junta de Andalucía), and by the ESF EUROCORES program EuroGIGA-ComPoSe IP04-MICINN Project EUI-EURC-201-4306. R. Fabila-Monroy was partially supported by Conacyt of Mexico, Grant 153984.

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Aichholzer, O., Caraballo, L.E., Díaz-Báñez, J.M. et al. Characterization of Extremal Antipodal Polygons. Graphs and Combinatorics 31, 321–333 (2015). https://doi.org/10.1007/s00373-015-1548-z

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  • DOI: https://doi.org/10.1007/s00373-015-1548-z

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