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Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem

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Abstract

The cyclohedron W n , known also as the Bott-Taubes polytope, arises both as the polyhedral realization of the poset of all cyclic bracketings of the word x 1 x 2...x n and as an essential part of the Fulton-MacPherson compactification of the configuration space of n distinct, labelled points on the circle S 1. The “polygonal pegs problem” asks whether every simple, closed curve in the plane or in the higher dimensional space admits an inscribed polygon of a given shape. We develop a new approach to the polygonal pegs problem based on the Fulton-MacPherson (Axelrod-Singer, Kontsevich) compactification of the configuration space of (cyclically) ordered n-element subsets in S 1. Among the results obtained by this method are proofs of Grünbaum’s conjecture about affine regular hexagons inscribed in smooth Jordan curves and a new proof of the conjecture of Hadwiger about inscribed parallelograms in smooth, simple, closed curves in the 3-space (originally established by Makeev in [Mak]).

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Authors and Affiliations

  1. Faculty of Mathematics, University of Belgrade, Belgrade, Yugoslavia

    Siniša T. Vrećica

  2. Mathematical Institute, SANU, Kneza Mihaila 36, 11001, Beogard, p.p. 367, Serbia

    Rade T. Živaljević

Authors
  1. Siniša T. Vrećica
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  2. Rade T. Živaljević
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Correspondence to Siniša T. Vrećica.

Additional information

To Anders Björner, on the occasion of his 60th birthday.

Supported by Grants 144014 and 144026 of the Serbian Ministry of Science and Technology.

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Vrećica, S.T., Živaljević, R.T. Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem. Isr. J. Math. 184, 221–249 (2011). https://doi.org/10.1007/s11856-011-0066-9

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  • DOI: https://doi.org/10.1007/s11856-011-0066-9

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