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

Abstract

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 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 Makeev in [Mak]).