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Metric entropy of homogeneous spaces. (English) Zbl 0927.46047
Alicki, Robert (ed.) et al., Quantum probability. Workshop, Gdańsk, Poland, July 1–6, 1997. Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 43, 395-410 (1998).
Summary: For a precompact subset \(K\) of a metric space and \(\varepsilon>0\), the covering number \(N(K,\varepsilon)\) is defined as the smallest number of balls of radius \(\varepsilon\) whose union covers \(K\). Knowledge of the 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 the author’s earlier results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. The argument uses a characterization of geodesics in \(U(n)\) (or \(\text{SO}(m)\)) for a class of non-Riemannian Finsler metric structures.
For the entire collection see [Zbl 0903.00097].

MSC:
46L53 Noncommutative probability and statistics
46L85 Noncommutative topology
53C30 Differential geometry of homogeneous manifolds
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C22 Geodesics in global differential geometry
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