椭圆多胞形与线性算子的不变范数
Elliptic polytopes and invariant norms of linear operators
摘要：
本文讨论了R^d中的椭圆多面体的构造问题，椭圆多面体是由许多具有公共中心的二维椭圆组成的凸包。这些集合出现在研究矩阵的谱性质、长矩阵乘积的渐近性、李雅普诺夫稳定性等方面。构造椭圆的主要问题是判断给定椭圆是否在其他椭圆的凸包内。通过考虑一个等价的优化问题，分析了该问题的计算复杂性。我们表明，该问题的局部极值的数量可能在d中呈指数增长。对于d= 2，3，它承认一个显式的解决方案，为任意数量的椭圆，对于更高的维度，几个几何方法的近似解。对这些方法进行了数值分析，并在应用中证明了它们的有效性。
英文摘要：
We address the problem of constructing elliptic polytopes in R^d, which are convex hulls of finitely many two-dimensional ellipses with a common center. Such sets arise in the study of spectral properties of matrices, asymptotics of long matrix products, in the Lyapunov stability, etc.. The main issue in the construction is to decide whether a given ellipse is in the convex hull of others. The computational complexity of this problem is analysed by considering an equivalent optimisation problem. We show that the number of local extrema of that problem may grow exponentially in d. For d=2,3, it admits an explicit solution for an arbitrary number of ellipses; for higher dimensions, several geometric methods for approximate solutions are derived. Those methods are analysed numerically and their efficiency is demonstrated in applications.
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。
