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

TL;DR

This paper introduces a novel approach to the polygonal pegs problem using Fulton-MacPherson compactification, leading to proofs of longstanding conjectures about inscribed polygons in smooth curves.

Contribution

It develops a new method based on Fulton-MacPherson compactification to address the polygonal pegs problem, proving conjectures by Grunbaum and Hadwiger.

Findings

Proof of Grunbaum's conjecture on affine regular hexagons

New proof of Hadwiger's conjecture on inscribed parallelograms in 3D

Application of Fulton-MacPherson compactification to geometric problems

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).