论文标题
Siegel磁盘的Hilbert几何学：Siegel-Klein磁盘模型
Hilbert geometry of the Siegel disk: The Siegel-Klein disk model
论文作者
论文摘要
我们研究了由Siegel Disk域引起的Hilbert几何形状，这是一组开放的操作员规范的复杂平方矩阵，严格少于一个。这种希尔伯特几何形状产生了双曲几何形状的klein磁盘模型的概括，因此称为siegel-klein磁盘模型，可以用经典的西格尔上层平面和磁盘域区分它。在Siegel-Klein磁盘中，大地测量是通过构造始终独特而欧几里得直的，可以从计算几何形状设计有效的几何算法和数据结构。 For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel-Poincaré disk and in the Siegel-Klein disk: We demonstrate that geometric computing in the Siegel-Klein disk allows一个（i）绕过Siegel-Poincaré磁盘模型中BC算法的每次迭代所需的磁盘起源的一（i），以及（ii）以近似于Siegel-klein距离，并保证了从嵌套的Hilbert hilbertbert Geometries衍生而来的Siegel-Klein距离。
We study the Hilbert geometry induced by the Siegel disk domain, an open bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel-Klein disk model to differentiate it with the classical Siegel upper plane and disk domains. In the Siegel-Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data-structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel-Poincaré disk and in the Siegel-Klein disk: We demonstrate that geometric computing in the Siegel-Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel-Poincaré disk model, and (ii) to approximate fast and numerically the Siegel-Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.
