Duality of metric entropy. (English) Zbl 1072.52001

The covering number, \(N(K,T)\), of convex bodies \(K,T\subset{\mathbb R}^n\), is the smallest number of translates of \(T\) needed to cover \(K\). Let \(B\) denote the Euclidean unit ball in \({\mathbb R}^n\).
The main result is the following. There exist constants \(\alpha, \beta>0\) such that for any convex body \(K\subset{\mathbb R}^n\) with \(K=-K\), we have \(N(B,\alpha^{-1} K^\circ)^{1/\beta}\leq N(K,B)\leq N(B,\alpha K^\circ)^\beta\), where \(K^\circ\) is the polar of \(K\). The constants \(\alpha, \beta\) are independent of \(n\) and \(K\). The technique of this paper gives \(\beta=2+\varepsilon\) for any \(\varepsilon > 0\).
This solves a special case of a conjecture of Pietsch on entropy numbers of operators and their duals. The introduction of the paper discusses the relationship between the main result and entropy numbers, in particular the conjecture of Pietsch, as well as the uses of covering numbers in information theory and the theory of Gaussian processes.


52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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