  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).
