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

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dc.creatorVrecica, Sinisa
dc.creatorZivaljevic, Rade
dc.date2008-10-08
dc.date2008-11-11
dc.date.accessioned2026-07-07T10:17:01Z
dc.date.available2026-07-07T10:17:01Z
dc.descriptionThe cyclohedron (Bott-Taubes polytope) arises both as the polyhedral realization of the poset of all cyclic bracketings of a circular word 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 Grunbaum'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 Victor Makeev).
dc.descriptionWe include a reference to a paper of Victor Makeev and acknowledge his priority in proving our Theorem 11 (related to Hadwiger's conjecture)
dc.identifierhttps://arxiv.org/abs/0810.1439
dc.identifierhttp://arxiv.org/abs/0810.1439
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/173710
dc.subjectCombinatorics
dc.subjectAlgebraic Topology
dc.subject53A04, 54H25, 55M20, 55R80
dc.titleFulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem
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