Duality of metric entropy in Euclidean space

Shiri Artstein; Vitali D. Milman; Stanislaw J. Szarek

doi:10.1016/j.crma.2003.09.033

Functional Analysis/Geometry

Duality of metric entropy in Euclidean space

Presented by: Mikhaël Gromov

Shiri Artstein ¹; Vitali D. Milman ¹; Stanislaw J. Szarek ^2,³

¹ School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
² Équipe d'analyse fonctionnelle, BC 186, Université Paris VI, 4, place Jussieu, 75252 Paris, France
³ Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106-7058, USA

Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 711-714.

Abstract
Résumé

Let K be a convex body in a Euclidean space, K° its polar body and D the Euclidean unit ball. We prove that the covering numbers N(K,tD) and N(D,tK°) are comparable in the appropriate sense, uniformly over symmetric convex bodies K, over t>0 and over the dimension of the space. 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.

Soit K un corps convexe d'un espace euclidien. Nous notons K° le polaire de K et D la boule unité euclidienne. Nous montrons que les nombres de recouvrement N(K,tD) et N(D,tK°) sont équivalents dans un sens approprié, uniformément sur tous les corps convexes symétriques, pour t>0 et pour toute dimension d'espace. En particulier, nous confirmons la conjecture concernant la dualité des nombres d'entropie des opérateurs compacts entre espaces de Banach, conjecture formulée par Pietsch en 1972 dans le cas fondamental où l'un des espaces est hilbertien.

Received: 2003-08-13
Accepted: 2003-09-30
Published online: 2003-11-07

DOI: 10.1016/j.crma.2003.09.033

Author's affiliations:

Shiri Artstein ¹; Vitali D. Milman ¹; Stanislaw J. Szarek ^{2, 3}

¹ School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
² Équipe d'analyse fonctionnelle, BC 186, Université Paris VI, 4, place Jussieu, 75252 Paris, France
³ Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106-7058, USA

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     author = {Shiri Artstein and Vitali D. Milman and Stanislaw J. Szarek},
     title = {Duality of metric entropy in {Euclidean} space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {711--714},
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     number = {11},
     year = {2003},
     doi = {10.1016/j.crma.2003.09.033},
     language = {en},
}

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Shiri Artstein; Vitali D. Milman; Stanislaw J. Szarek. Duality of metric entropy in Euclidean space. Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 711-714. doi : 10.1016/j.crma.2003.09.033. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.033/

References
Cited by

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[2] S. Artstein; V.D. Milman; S.J. Szarek More on the duality conjecture for entropy numbers, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003) no. 6, pp. 479-482

[3] H. König; V. Milman On the covering numbers of convex bodies, Geometric Aspects of Functional Analysis (1985–1986), Lecture Notes in Math., 1267, Springer, Berlin, 1987, pp. 82-95

[4] M. Ledoux Isoperimetry and Gaussian analysis, Lectures on Probability Theory and Statistics (Saint-Flour, 1994), Lecture Notes in Math., 1648, Springer, Berlin, 1996, pp. 165-294

[5] V.D. Milman; S.J. Szarek A geometric lemma and duality of entropy numbers, Geometric Aspects of Functional Analysis (1996–2000), Lecture Notes in Math., 1745, Springer, Berlin, 2000, pp. 191-222

[6] A. Pietsch Theorie der Operatorenideale (Zusammenfassung), Friedrich-Schiller-Universität Jena, 1972

[7] N. Tomczak-Jaegermann Dualité des nombres d'entropie pour des opérateurs à valeurs dans un espace de Hilbert, C. R. Acad. Sci. Paris, Ser. I, Volume 305 (1987) no. 7, pp. 299-301

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