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

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 FultonMacPherson (Axelrod-Singer, Kontsevich) compactification of the configuration space of (cyclically) ordered n-element subsets in S1. Among the new results obtained by this method are proofs of Grünbaum’s conjecture about inscribed affine regular hexagons in smooth Jordan curves and the conjecture of Hadwiger about inscribed parallelograms in smooth, simple, closed curves in the 3-space.