Comptes Rendus
no. 6
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

Shiri Artstein1 ; Vitali D. Milman2 ; Stanislaw J. Szarek2, 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, Volume 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
Shiri Artstein 1 ; Vitali D. Milman 2 ; Stanislaw J. Szarek 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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Shiri Artstein; Vitali D. Milman; Stanislaw J. Szarek. More on the duality conjecture for entropy numbers. Comptes Rendus. Mathématique, Volume 336 (2003) no. 6, pp. 479-482. doi : 10.1016/S1631-073X(03)00102-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00102-X/

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

[2] J. Bourgain; A. Pajor; S.J. Szarek; N. Tomczak-Jaegermann 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] W.B. Johnson; J. Lindenstrauss 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] H. König; V. Milman 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] V.D. Milman 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] V.D. Milman; A. Pajor Entropy and asymptotic geometry of non-symmetric convex bodies, Adv. in Math., Volume 152 (2000) no. 2, pp. 314-335

[7] 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

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

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

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

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

[12] N. Tomczak-Jaegermann 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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