Geometry/Functional Analysis
More on the duality conjecture for entropy numbers
[Sur la conjecture de la dualité pour les nombres d'entropie]

Présenté par : Michel Talagrand

Artstein, Shiri1 ; Milman, Vitali D.2 ; Szarek, Stanislaw J.2, 3
1 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
2 Équipe d'analyse fonctionnelle, B.C. 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, Tome 336 (2003) no. 6, pp. 479-482.

Nous démontrons, à un facteur logarithmique près, la conjecture concernant la dualité de nombres d'entropie dans le cas où l'un de deux corps est un ellipsoı̈de.

We verify, up to a logarithmic factor, the duality conjecture for entropy numbers in the case where one of the bodies is an ellipsoid.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00102-X
Artstein, Shiri 1 ; Milman, Vitali D. 2 ; Szarek, Stanislaw J. 2, 3

1 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
2 Équipe d'analyse fonctionnelle, B.C. 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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Artstein, Shiri; Milman, Vitali D.; Szarek, Stanislaw J. More on the duality conjecture for entropy numbers. Comptes Rendus. Mathématique, Tome 336 (2003) no. 6, pp. 479-482. doi : 10.1016/S1631-073X(03)00102-X. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00102-X/

[1] Artstein, S. Proportional concentration phenomena, Israel J. Math., Volume 132 (2002), pp. 337-358

[2] Bourgain, J.; Pajor, A.; Szarek, S.J.; Tomczak-Jaegermann, N. On the duality problem for entropy numbers of operators, Geometric Aspects of Functional Analysis (1987–88), Lecture Notes in Math., 1376, Springer, Berlin, 1989, pp. 50-63

[3] Johnson, W.B.; Lindenstrauss, J. Extensions of Lipschitz mappings into a Hilbert space, Conference in Modern Analysis and Probability, New Haven, CO, Contemp. Math., 26, American Mathematical Society, Providence, RI, 1982, pp. 189-206

[4] König, H.; Milman, V. On the covering numbers of convex bodies, Geometric Aspects of Functional Analysis (1985–86), Lecture Notes in Math., 1267, Springer, Berlin, 1987, pp. 82-95

[5] Milman, V.D. A note on a low M * -estimate, Geometry of Banach Spaces, Strobl, 1989, London Math. Soc. Lecture Note Ser., 158, Cambridge University Press, Cambridge, 1990, pp. 219-229

[6] Milman, V.D.; Pajor, A. Entropy and asymptotic geometry of non-symmetric convex bodies, Adv. in Math., Volume 152 (2000) no. 2, pp. 314-335

[7] Milman, V.D.; Szarek, S.J. 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

[8] Milman, V.D.; Szarek, S.J. A geometric approach to duality of metric entropy, C. R. Acad. Sci. Paris, Sér. I, Volume 332 (2001) no. 2, pp. 157-162

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

[10] Pisier, G. A new approach to several results of V. Milman, J. Reine Angew. Math., Volume 393 (1989), pp. 115-131

[11] Pisier, G. The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math., 94, Cambridge University Press, Cambridge, 1989

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

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