双曲细胞的生长

摘要：

考虑单位圆盘$\{z:\, |z|<1\}$中的双曲多边形，其顶点都在单位圆$\{z:\, |z|=1\}$上，当每个$n$边形通过在其所有边上反转生成一个$n(n-1)$边形时，这类多边形的增长过程.本文证明了凸双曲$n$ -边形反演的一些一般单调性结果，并解决了一个极值问题：在所有包含原点的凸双曲$4$ -边形中，给定的双曲$4$-边形的最长边的反演边长对于正则双曲$4$-边形是最小的.

英文摘要：

We consider a hyperbolic polygon in the unit disk $\{z:\, |z|<1\}$ with all its vertices on the unit circle $\{z:\, |z|=1\}$ and a growth process of such polygons when each $n$-gon generates an $n(n-1)$-gon by inverting itself across all of its sides. In this paper, we prove some general monotonicity results of inversion for convex hyperbolic $n$-gons and solve an extremal problem that, among all convex hyperbolic $4$-gons containing the origin, the inverted side length of the longest side of the given hyperbolic $4$-gon is minimal for the regular hyperbolic $4$-gon.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。