论文标题
恒定曲率空间中的离散等级问题
Discrete isoperimetric problems in spaces of constant curvature
论文作者
论文摘要
本文的目的是证明简单和多台面的等距不平等现象，在欧几里得，球形和双曲线$ d $ -space中具有$ d+2 $的顶点。特别是，我们发现给定的Inradius的最小体积$ d $二维的双曲线简单和球形四面体。此外，我们研究了带有给定圆周的$ d+2 $顶点的最大体积球形和双曲线多型的特性，以及带有$ d+2 $顶点的双曲线多型，具有给定的inradius，并且具有最小的体积或最小的总边缘。最后，对于任何$ 1 \ leq k \ leq d $，我们使用固定的inradius和最少的$ k $ skeleton的欧几里德简单和多面体的属性。我们研究的主要工具是欧几里得，球形和双曲线施泰纳对称性。
The aim of this paper is to prove isoperimetric inequalities for simplices and polytopes with $d+2$ vertices in Euclidean, spherical and hyperbolic $d$-space. In particular, we find the minimal volume $d$-dimensional hyperbolic simplices and spherical tetrahedra of a given inradius. Furthermore, we investigate the properties of maximal volume spherical and hyperbolic polytopes with $d+2$ vertices with a given circumradius, and the hyperbolic polytopes with $d+2$ vertices with a given inradius and having a minimal volume or minimal total edge length. Finally, for any $1 \leq k \leq d$, we investigate the properties of Euclidean simplices and polytopes with $d+2$ vertices having a fixed inradius and a minimal volume of its $k$-skeleton. The main tool of our investigation is Euclidean, spherical and hyperbolic Steiner symmetrization.
