Authors: Vrećica, Siniša
Živaljević, Rade 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem
Journal: Israel Journal of Mathematics
Volume: 184
Issue: 1
First page: 221
Last page: 249
Issue Date: 1-Aug-2011
Rank: M21
ISSN: 0021-2172
DOI: 10.1007/s11856-011-0066-9
The cyclohedron Wn, known also as the Bott-Taubes polytope, arises both as the polyhedral realization of the poset of all cyclic bracketings of the word x1x2··· xn and as an essential part of the Fulton-MacPherson compactification of the configuration space of n distinct, labelled points on the circle S1. 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 S1. 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]).
Publisher: Springer Link
Project: Supported by Grants 144014 and 144026 of the Serbian Ministry of Science and Technology

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