齐性空间的度量熵与经典李群的Finsler几何
Metric entropy of homogeneous spaces and Finsler geometry of classical Lie groups
摘要：
对于度量空间的（紧）子集$K$和$\varepsilon > 0$，{ \em覆盖数} $N(K , \varepsilon )$定义为半径为$\varepsilon$且其并集覆盖$K$的球的最小数目。{ \em度量熵}的知识，即，度量空间（族）的覆盖数的渐近行为在许多数学领域（几何、泛函分析、概率、编码理论等）中是重要的。本文给出了一大类酉（或正交）群的齐性空间关于某些自然度量的覆盖数的渐近正确估计，特别是由算子范数诱导的覆盖数.这推广了早先作者关于格拉斯曼流形的覆盖数的结果;推广的动机是应用到非交换概率和算子代数。在这个过程中，我们给出了一类非黎曼度量结构的$U(n)$（或$SO(m)$）中测地线的一个特征。
英文摘要：
For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em covering number} $N(K , \varepsilon )$ is defined as the smallest number of balls of radius $\varepsilon$ whose union covers $K$. Knowledge of the {\em metric entropy}, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous spaces of unitary (or orthogonal) groups with respect to some natural metrics, most notably the one induced by the operator norm. This generalizes earlier author's results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. In the process we give a characterization of geodesics in $U(n)$ (or $SO(m)$) for a class of non-Riemannian metric structures.
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