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Symmetric curves, hexagons, and the girth of spheres in dimension 3

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Abstract

It is proved that every centrally symmetric simple closed curve on the boundary of a centrally symmetric convex body in a three-dimensional linear space possesses an inscribed concentric affinely regular hexagon. This result is used to settle affirmatively a conjecture in [2] about the metric structure of the unit spheres of three-dimensional normed space.

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References

  1. H. Cartan,Théorie des faisceaux. Séminaire H. Cartan, 3e année (1950/1951). Éc. Nat. Supér., Paris. Exposé 20.

  2. J. J. Schäffer,Inner diameter, perimeter, and girth of spheres, Math. Ann.173 (1967), 59–79.

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  3. ——,Addendum: Inner diameter, perimeter, and girth of spheres, Math. Ann.173 (1967), 79–82.

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Authors and Affiliations

  1. Universidad de la República., Montevideo, Uruguay

    Juan Jorge Schäffer

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  1. Juan Jorge Schäffer
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Schäffer, J.J. Symmetric curves, hexagons, and the girth of spheres in dimension 3. Israel J. Math. 6, 202–205 (1968). https://doi.org/10.1007/BF02760252

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  • DOI: https://doi.org/10.1007/BF02760252

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