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For two convex bodies $K$ and $T$ in $\mathbb{R}^n$, 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 $K^{\circ}$ the polar body of $K$ and by $D$ the euclidean unit ball in $\mathbb{R}^n$. We prove that the two functions of $t$, $N(K,tD)$ and $N(D, tK^{\circ})$, are equivalent in the appropriate sense, uniformly over symmetric convex bodies $K \subset \mathbb{R}^n$ and over $n \in \mathbb{N}$. 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.
Shiri Artstein 1 ; Vitali Milman 1 ; Stanisław J. Szarek 2
@article{10_4007_annals_2004_159_1313,
author = {Shiri Artstein and Vitali Milman and Stanis{\l}aw J. Szarek},
title = {Duality of metric entropy},
journal = {Annals of mathematics},
pages = {1313--1328},
publisher = {mathdoc},
volume = {159},
number = {3},
year = {2004},
doi = {10.4007/annals.2004.159.1313},
mrnumber = {2113023},
zbl = {1072.52001},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.159.1313/}
}
TY - JOUR AU - Shiri Artstein AU - Vitali Milman AU - Stanisław J. Szarek TI - Duality of metric entropy JO - Annals of mathematics PY - 2004 SP - 1313 EP - 1328 VL - 159 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.159.1313/ DO - 10.4007/annals.2004.159.1313 LA - en ID - 10_4007_annals_2004_159_1313 ER -
%0 Journal Article %A Shiri Artstein %A Vitali Milman %A Stanisław J. Szarek %T Duality of metric entropy %J Annals of mathematics %D 2004 %P 1313-1328 %V 159 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.159.1313/ %R 10.4007/annals.2004.159.1313 %G en %F 10_4007_annals_2004_159_1313
Shiri Artstein; Vitali Milman; Stanisław J. Szarek. Duality of metric entropy. Annals of mathematics, Tome 159 (2004) no. 3, pp. 1313-1328. doi: 10.4007/annals.2004.159.1313
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