Fulton-MacPherson紧化、多面体和多边形钉问题
Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem
摘要：
圆多面体（Bott-Taubes多面体）既是圆字的所有循环括号的偏序集的多面体实现，也是圆S ^[1]上n个不同标号点的位形空间的富尔顿-麦克弗森紧化的一个基本部分。“多边形桩问题”的问题是，是否每一个简单的，闭曲线在平面上或在更高的维空间允许一个给定形状的内接多边形。基于S^1中（循环）有序n元子集的配置空间的Fulton-MacPherson（Axelrod-Singer，Kontsevich）紧化，我们发展了一种新的方法来解决多边形钉问题。其中所获得的结果是证明Grunbaum猜想的仿射正六边形内接光滑约旦曲线和一个新的证明Hadwiger猜想内接平行四边形光滑，简单，闭曲线在3-空间（最初建立的维克托Makeev）。
英文摘要：
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).
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