Randomized isomorphic Dvoretzky theorem
Comptes Rendus. Mathématique, Volume 335 (2002) no. 4, pp. 345-350.

Let K be a symmetric convex body in N for which B2N is the ellipsoid of minimal volume. We provide estimates for the geometric distance of a ‘typical’ rank n projection of K to B2n, for 1⩽n<N. Known examples show that the resulting estimates are optimal (up to numerical constants) even for the Banach–Mazur distance.

Soit K un corps convexe symétrique de N dont l'ellipsoı̈de de volume minimal le contenant est la boule euclidienne B2N. Nous estimons la distance géométrique de projections « typiques » de rang n de K à la boule B2n pour tout n∈{1,…,N−1} (i.e. nous prouvons qu'il en existe une grande proportion au sens de la mesure de Haar normalisée sur la grassmanienne). Des exemples bien connus permettent de dire que ces estimations sont optimales (à des constantes numériques près), même pour la distance de Banach–Mazur.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(02)02476-7
Litvak, Alexander 1; Mankiewicz, Piotr 2; Tomczak-Jaegermann, Nicole 1

1 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
2 Institute of Mathematics, Polish Academy of Sciences, PB 137, 00-950 Warsaw, Poland
@article{CRMATH_2002__335_4_345_0,
     author = {Litvak, Alexander and Mankiewicz, Piotr and Tomczak-Jaegermann, Nicole},
     title = {Randomized isomorphic {Dvoretzky} theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {345--350},
     publisher = {Elsevier},
     volume = {335},
     number = {4},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02476-7},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02476-7/}
}
TY  - JOUR
AU  - Litvak, Alexander
AU  - Mankiewicz, Piotr
AU  - Tomczak-Jaegermann, Nicole
TI  - Randomized isomorphic Dvoretzky theorem
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 345
EP  - 350
VL  - 335
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(02)02476-7/
DO  - 10.1016/S1631-073X(02)02476-7
LA  - en
ID  - CRMATH_2002__335_4_345_0
ER  - 
%0 Journal Article
%A Litvak, Alexander
%A Mankiewicz, Piotr
%A Tomczak-Jaegermann, Nicole
%T Randomized isomorphic Dvoretzky theorem
%J Comptes Rendus. Mathématique
%D 2002
%P 345-350
%V 335
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(02)02476-7/
%R 10.1016/S1631-073X(02)02476-7
%G en
%F CRMATH_2002__335_4_345_0
Litvak, Alexander; Mankiewicz, Piotr; Tomczak-Jaegermann, Nicole. Randomized isomorphic Dvoretzky theorem. Comptes Rendus. Mathématique, Volume 335 (2002) no. 4, pp. 345-350. doi : 10.1016/S1631-073X(02)02476-7. http://www.numdam.org/articles/10.1016/S1631-073X(02)02476-7/

[1] Gluskin, E.D. Extremal properties of orthogonal parallelopipeds and their applications to the geometry of Banacj spaces, Math. USSR-Sb., Volume 64 (1989), pp. 85-96

[2] Gordon, Y. Some inequalities for Gaussian processes and applications, Israel J. Math., Volume 50 (1985), pp. 265-289

[3] Gordon, Y.; Guédon, O.; Meyer, M. An isomorphic Dvoretzky theorem for convex bodies, Studia Math., Volume 127 (1998), pp. 191-200

[4] Guédon, O. Gaussian version of a theorem of Milman and Schechtman, Positivity, Volume 1 (1997), pp. 1-5

[5] Lifshits, M.A. Gaussian Random Functions, Math. Appl., 322, Kluwer Academic, Dordrecht, 1995

[6] Litvak, A.E.; Tomczak-Jaegermann, N. Random aspects of high dimensional convex bodies, Geometric Aspects of Functional Analysis, Israeli Seminar, Lecture Notes in Math., 1795, Springer, 2000, pp. 169-191

[7] Mankiewicz, P.; Tomczak-Jaegermann, N. Geometry of families of random projections of symmetric convex bodies, Geom. Funct. Anal., Volume 11 (2001), pp. 1282-1326

[8] Milman, V.D. A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Funct. Anal. Appl., Volume 5 (1971), pp. 28-37 (English translation)

[9] Milman, V.D. Surprising geometric phenomena in high dimensional convexity theory, Proceedings of II European Congress of Mathematicians, Budapest, 1996

[10] Milman, V.D.; Schechtman, G. Finite-Dimensional Normed Spaces, Lecture Notes in Math., 1200, Springer, Berlin, 1986

[11] Milman, V.D.; Schechtman, G. An ‘isomorphic’ version of Dvoretzky's theorem, C. R. Acad. Sci. Paris, Série I, Volume 321 (1995), pp. 541-544

[12] Milman, V.D.; Schechtman, G. Global versus local asymptotic theories of finite-dimensional normed spaces, Duke Math. J., Volume 90 (1997), pp. 73-93

[13] Milman, V.D.; Schechtman, G. An ‘Isomorphic’ Version of Dvoretzky's Theorem, II; Convex Geometric Analysis, MSRI Publ., 34, Cambridge University Press, 1998 (pp. 159–164)

[14] Pisier, G. The Volume of Convex Bodies and Banach Space Geometry, Cambridge University Press, 1989

[15] Vershynin, R. John's decomposition: selecting a large part, Israel J. Math., Volume 122 (2001), pp. 253-277

Cited by Sources: