Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem
Abstract
The 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).
Cite
@article{arxiv.0810.1439,
title = {Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem},
author = {Sinisa Vrecica and Rade Zivaljevic},
journal= {arXiv preprint arXiv:0810.1439},
year = {2008}
}
Comments
We include a reference to a paper of Victor Makeev and acknowledge his priority in proving our Theorem 11 (related to Hadwiger's conjecture)