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Envelopes of \(\alpha \)-Sections

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Convexity and Discrete Geometry Including Graph Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 148))

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Abstract

Let K be a planar convex body af area |K|, and take \(0<\alpha <1\). An \(\alpha \)-section of K is a line cutting K into two parts, one of which has area \(\alpha |K|\). This article presents a systematic study of the envelope of \(\alpha \)-sections and its dependence on \(\alpha \). Several open questions are asked, one of them in relation to a problem of fair partitioning.

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Notes

  1. 1.

    In fact (12) is stated for \(\alpha \) fixed, but its proof uses only (3) and (10)—which can easily be adapted to our situation—and the continuity of the function \(\mathrm{cotan}\).

  2. 2.

    A classical conjecture states that only ellipses have a dual billiard map which is integrable.

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Acknowledgments

The authors thank Theodor Hangan and Tudor Zamfirescu for fruitful discussions and for having pointed to them several references. They are also indebted to the anonymous referee who drew to their attention the reference [21]. The third author thanks the Universitè de Haute Alsace for a one-month grant and its hospitality, and acknowledges partial support from the Roumanian National Authority for Scientific Research, CNCS-UEFISCDI, grant PN-II-ID-PCE-2011-3-0533.

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Informatique et Applications, Faculté des Sciences et Techniques Université de Haute Alsace, 2 rue des Frères Lumière, 68093, Mulhouse Cedex, France

    Nicolas Chevallier & Augustin Fruchard

  2. Simion Stoilow Institute of Mathematics of the Roumanian Academy, P.O. Box 1-764, 70700, Bucharest, Romania

    Costin Vîlcu

Authors
  1. Nicolas Chevallier
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  2. Augustin Fruchard
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  3. Costin Vîlcu
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Corresponding author

Correspondence to Augustin Fruchard .

Editor information

Editors and Affiliations

  1. Einstein Institute for Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

    Karim Adiprasito

  2. Hungarian Academy of Sciences, Rényi Institute of Mathematics, Budapest, Hungary

    Imre Bárány

  3. Mathematics of the Roumanian Academy, ``Simion Stoilow'' Institute of, Bucharest, Romania

    Costin Vilcu

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Chevallier, N., Fruchard, A., Vîlcu, C. (2016). Envelopes of \(\alpha \)-Sections. In: Adiprasito, K., Bárány, I., Vilcu, C. (eds) Convexity and Discrete Geometry Including Graph Theory. Springer Proceedings in Mathematics & Statistics, vol 148. Springer, Cham. https://doi.org/10.1007/978-3-319-28186-5_17

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