随机滚动多面体的轨迹
On traces of randomly rolling polytopes
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中文总结 AI 辅助
该研究针对三维凸多面体随机滚动的轨迹问题，证明若其轨迹有收敛子序列，则顶点可达点集几乎必然在平面处处稠密，解决了Hegyvári的猜想。
中文摘要 AI 辅助
设𝒫为一个三维凸多面体，其某一面放置于平面上。每一步，𝒫可绕当前位于平面上的面的随机选定边滚动，直至相邻面静止于平面上。𝒫的轨迹是从固定初始位置出发，通过有限次滚动，其顶点可到达的所有平面点的集合。我们证明：若𝒫的轨迹存在收敛子序列，则几乎必然，随机滚动的𝒫的顶点可到达的点集在平面上处处稠密。这解决了Hegyvári的一个猜想。
英文摘要
Let $\mathcal{P}$ be a three-dimensional convex polytope resting with one of its faces on the plane. At each step, $\mathcal{P}$ is allowed to roll over a randomly selected edge of the face currently lying on the plane, until the adjacent face comes to rest on the plane. The trace of $\mathcal{P}$ is the set of all points of the plane that can be reached by a vertex of $\mathcal{P}$, starting from a fixed initial position and performing a finite sequence of rolls. We prove that if the trace of $\mathcal{P}$ has a convergent subsequence, then, with probability one, the set of points reached by the vertices of a randomly rolling copy of $\mathcal{P}$ is everywhere dense in the plane. This settles a conjecture of Hegyvári.
