AI 中文总结
研究循环标记平面\(n\)顶点配置通过正顶点 - 质心移动变换为正\(n\)边形的问题，给出移动条件及更一般情况，还涉及\(\mathbb{R}^d\)中配置可达性，通过特定代数方法和论证确定群，实现精确可达性。
AI 中文摘要
我们在总质心处选定顶点不动的自然约定下解决了开放问题项目的问题60。一个循环标记的平面\(n\)顶点配置（\(n\geq3\)），当且仅当它不共线时，可通过有限次顶点 - 质心移动精确变换为正\(n\)边形。肯定方向与该约定无关，且所有移动可设为正，选定顶点从不穿过其他顶点的质心。更一般地，仿射生成配置空间的连通开子集中的任意两个配置可通过正移动连接，且整个连续执行过程保持在该子集中。对于简单标记\(n\)边形，允许直角顶点，这在每个方向类中精确保持简单性的同时实现精确可达性。在\(\mathbb{R}^d\)中，当\(n\geq d + 2\)时所有全维标记\(n\)点配置相互可达，当\(n = d + 1\)时方向是唯一障碍。将配置写为坐标矩阵，每次移动成为单位矩阵的秩一扰动。明确的单步移动曲线和三步共轭词产生在每个仿射生成配置处具有可逆微分的有限词端点映射。逆函数定理给出精确的局部可达性，所有中间状态局限于规定邻域，连通性使其成为全局可达性。一个基本的耳缩减论证确立了简单多边形方向类所需的连通性。相同代数精确确定了由正移动生成的群。证明是存在性的且非定量的。
英文摘要
We resolve Problem 60 of The Open Problems Project under the natural convention that a selected vertex at the total centroid is immobile. A cyclically labeled planar $n$-vertex configuration, $n\ge3$, can be transformed exactly into a regular $n$-gon by finitely many vertex-centroid moves if and only if it is noncollinear. The affirmative direction is independent of this convention, and all moves may be chosen positive, so the selected vertex never crosses the centroid of the other vertices. More generally, any two configurations in a connected open subset of the affinely spanning configuration space can be joined by positive moves whose entire continuous execution remains in that subset. For simple labeled $n$-gons, allowing straight-angle vertices, this yields exact reachability while preserving simplicity precisely within each orientation class. In $\mathbb{R}^d$, it yields mutual reachability of all full-dimensional labeled $n$-point configurations when $n\ge d+2$, while orientation is the only obstruction when $n=d+1$. Writing configurations as coordinate matrices, each move becomes a rank-one perturbation of the identity. Explicit one-move curves and three-move conjugation words yield a finite-word endpoint map with invertible differential at every affinely spanning configuration. The inverse function theorem gives exact local reachability with all intermediate states confined to a prescribed neighborhood, and connectedness makes it global. An elementary ear-reduction argument establishes the required connectivity of simple-polygon orientation classes. The same algebra determines exactly the group generated by positive moves. The proof is existential and nonquantitative.
Comments16 pages, 3 figures
