Duality of metric entropy

Details Duality of metric entropy Duality of metric entropy

2004-07-14

arxiv.org/abs/math/0407236v1

Annals of Math. 159 (2004), no. 3, 1313-1328.

Mathematics

Functional Analysis

17 p., LATEX

Scientific paper

For two convex bodies K and T in $R^n$, the covering number of K by T, denoted N(K,T), is defined as the minimal number of translates of T needed to cover K. Let us denote by $K^o$ the polar body of K and by D the euclidean unit ball in $R^n$. We prove that the two functions of t, N(K, tD) and N(D, tK^o), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K in $R^n$ and over positive integers n. In particular, this verifies the duality conjecture for entropy numbers of linear operators, posed by Pietsch in 1972, in the central case when either the domain or the range of the operator is a Hilbert space.

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