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Metric Entropy of Homogeneous Spaces

Stanisław Szarek — 1998

Banach Center Publications

For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε 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...

On the geometry of proportional quotients of l ₁ m

Piotr Mankiewicz;  Stanisław J. Szarek — 2003

Studia Mathematica

We compare various constructions of random proportional quotients of l ₁ m (i.e., with the dimension of the quotient roughly equal to a fixed proportion of m as m → ∞) and show that several of those constructions are equivalent. As a consequence of our approach we conclude that the most natural “geometric” models possess a number of asymptotically extremal properties, some of which were hitherto not known for any model.

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