发表机构
Université Paris 13; Sun Yat-sen University（巴黎第十三大学; 中山大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了顶点足够多的凸多边形或含一个凹顶点的简单多边形，可放置单个穿孔使对应翻转图子图非强凸，完善了翻转图强凸性的研究。
AI 中文摘要
简单多边形P的顶点集X的三角剖分集合可构成翻转图$\u2139(P,X)$，其边连接仅差一条弧的两个三角剖分。翻转图的几何已被深入研究，已知当P为凸多边形、X不含穿孔（P内部的点）且至多含一个平顶点（边内部的点）时，含给定弧$\u03b5$的三角剖分构成的子图$\u2139_\u03b5(P,X)$在$\u2139(P,X)$中是强凸的；当X含至少两个穿孔或平顶点时，该强凸性不成立。本文解决了最后一个开放情况：对任意顶点足够多的凸多边形，总能在X中放置一个穿孔，使得$\u2139_\u03b5(P,X)$在$\u2139(P,X)$中不是强凸的；对含一个凹顶点的简单多边形，也证明了类似结果。证明的主要依据是一类三维三角剖分的分解引理，以及它们嵌入$\u210d^3$的双曲体积论证。
英文摘要
The set of the triangulations with vertex set $X$ of a simple polygon $\mathrm{P}$ can be structured into a flip-graph $\mathcal{F}(\mathrm{P},X)$ whose edges connect two triangulations that differ by a single arc. The geometry of flip-graphs has been thoroughly studied and it is known that the subgraph $\mathcal{F}_\varepsilon(\mathrm{P},X)$ induced by the triangulations that contain a given arc $\varepsilon$ is strongly convex in $\mathcal{F}(\mathrm{P},X)$ when $\mathrm{P}$ is convex and $X$ contains no puncture (points in the interior of $\mathrm{P}$) and at most one flat vertex (points in the interior of an edge). When $X$ contains at least two punctures or flat vertices, it is also known that this strong convexity property fails. Here, we close the last open case by showing that, for any convex polygon with sufficiently many vertices, one can always place a single puncture in $X$ in such a way that $\mathcal{F}_\varepsilon(\mathrm{P},X)$ is not strongly convex in $\mathcal{F}(\mathrm{P},X)$. We prove a similar result for simple polygons with a single reflex vertex. The main ingredients in our proofs are a decomposition lemma for a class of $3$-dimensional triangulations and a hyperbolic volume argument regarding their embedding into $\mathbb{H}^3$.
Comments38 pages, 14 figures
