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On a Helly-type question for central symmetry

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Abstract

We study a certain Helly-type question by Konrad Swanepoel. Assume that X is a set of points such that every k-subset of X is in centrally symmetric convex position, is it true that X must also be in centrally symmetric convex position? It is easy to see that this is false if \(k\le 5\), but it may be true for sufficiently large k. We show that the statement is not true even when \(k=8\), but \(k=6\) is enough if X is a simple closed curve.

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References

  1. D. Djukić, V. Janković, I. Matić, N. Petrović, in The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959–2009 Second Edition (Springer Science & Business Media, New York, 2011)

  2. J. Eckhoff, in Helly, Radon, and Carathéodory type theorems, Handbook of convex geometry (1993), p. 389–448

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  4. M.V. Smurov, in Problem 10.7, Russian Mathematical Olympiad (1996). http://www.imomath.com/index.php?options=Rus&mod=23&ttn=Russia

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Acknowledgements

The authors are thankful to Konrad Swanepoel for the interesting questions. We are also thankful to Imre Bárány and Jesús Jerónimo for many fruitful discussions while this work was in progress.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Brownsville, TX, USA

    Alexey Garber

  2. Centro de Ciencias Matemáticas, UNAM Campus Morelia, Morelia, Mexico

    Edgardo Roldán-Pensado

Authors
  1. Alexey Garber
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  2. Edgardo Roldán-Pensado
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Correspondence to Alexey Garber.

Additional information

Dedicated to Imre Bárány on his 70th birthday.

E. Roldán-Pensado: Supported by PAPIIT project IA102118.

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Garber, A., Roldán-Pensado, E. On a Helly-type question for central symmetry. Period Math Hung 79, 78–85 (2019). https://doi.org/10.1007/s10998-018-0263-y

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  • DOI: https://doi.org/10.1007/s10998-018-0263-y

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