Geometry/Functional Analysis
The Knaster problem and the geometry of high-dimensional cubes
Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 931-936.

We study questions of the following type: Given positive semi-definite matrix 𝒢, does there exist a sequence of vectors in  n whose Grammian equals to 𝒢 and which has some specified additional properties (typically related to the sup norm)? In particular, we show that the answer to the 1947 Knaster problem about real functions on spheres is negative for sufficiently large dimensions.

Nous étudions des questions du type suivant : Soit 𝒢 une matrice positive semi-définie, existe-t-il une suite de vecteurs dans n dont la matrice de Gram est égale à 𝒢 et qui possède certaines propriétés supplémentaires (typiquement liées à la norme sup) ? En particulier, nous montrons que la réponse au problème de Knaster datant de 1947 et concernant les fonctions réelles sur les sphères est négative en dimension suffisamment grande.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00226-7
Kashin, Boris S. 1; Szarek, Stanislaw J. 2, 3

1 Steklov Mathematical Institute, 8 Gubkina Street, 117966, GSP1, Moscow, Russia
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
@article{CRMATH_2003__336_11_931_0,
     author = {Kashin, Boris S. and Szarek, Stanislaw J.},
     title = {The {Knaster} problem and the geometry of high-dimensional cubes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {931--936},
     publisher = {Elsevier},
     volume = {336},
     number = {11},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00226-7},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00226-7/}
}
TY  - JOUR
AU  - Kashin, Boris S.
AU  - Szarek, Stanislaw J.
TI  - The Knaster problem and the geometry of high-dimensional cubes
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 931
EP  - 936
VL  - 336
IS  - 11
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(03)00226-7/
DO  - 10.1016/S1631-073X(03)00226-7
LA  - en
ID  - CRMATH_2003__336_11_931_0
ER  - 
%0 Journal Article
%A Kashin, Boris S.
%A Szarek, Stanislaw J.
%T The Knaster problem and the geometry of high-dimensional cubes
%J Comptes Rendus. Mathématique
%D 2003
%P 931-936
%V 336
%N 11
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00226-7/
%R 10.1016/S1631-073X(03)00226-7
%G en
%F CRMATH_2003__336_11_931_0
Kashin, Boris S.; Szarek, Stanislaw J. The Knaster problem and the geometry of high-dimensional cubes. Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 931-936. doi : 10.1016/S1631-073X(03)00226-7. http://www.numdam.org/articles/10.1016/S1631-073X(03)00226-7/

[1] Chen, W. Counterexamples to Knaster's conjecture, Topology, Volume 37 (1998) no. 2, pp. 401-405

[2] Floyd, E.E. Real-valued mappings of spheres, Proc. Amer. Math. Soc., Volume 6 (1955), pp. 957-959

[3] Gilbert, J.E.; Leih, T.J. Factorization, tensor products, and bilinear forms in Banach space theory, Notes in Banach Spaces, University Texas Press, Austin, TX, 1980, pp. 182-305

[4] Kakutani, S. A proof that there exists a circumscribing cube around any bounded closed convex set in 3 , Ann. of Math., Volume 43 (1942), pp. 739-741

[5] Kashin, B.S. The widths of certain finite-dimensional sets and classes of smooth functions, Izv. Akad. Nauk SSSR Ser. Mat., Volume 41 (1977), pp. 334-351 (in Russian)

[6] Knaster, B. Problem 4, Colloq. Math., Volume 30 (1947), pp. 30-31

[7] Makeev, V.V. Some properties of continuous mappings of spheres and problems in combinatorial geometry, Geometric Questions in the Theory of Functions and Sets, Kalinin. Gos. Univ, Kalinin, 1986, pp. 75-85 (in Russian)

[8] Megretski, A. Relaxations of quadratic programs in operator theory and system analysis, Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Oper. Theory Adv. Appl., 129, Birkhäuser, Basel, 2001, pp. 365-392

[9] Menshoff, D. Sur les séries de fonctions orthogonales bornées dans leur ensembles, Mat. Sb., Volume 3 (1938) no. 45, pp. 103-120

[10] Milman, V.D. A few observations on the connections between local theory and some other fields, Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math., 1317, Springer-Verlag, Berlin, 1988, pp. 283-289

[11] Olevskiı̌, A.M. Fourier Series with Respect to General Orthogonal Systems, Springer-Verlag, Berlin, 1975

[12] Pisier, G. Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math., 60, American Mathematical Society, Providence, RI, 1986

[13] Szarek, S.J. On Kashin's almost Euclidean orthogonal decomposition of ℓ1n, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., Volume 26 (1978), pp. 691-694

[14] Yamabe, H.; Yujobo, Z. On the continuous function defined on a sphere, Osaka Math. J., Volume 2 (1950) no. 1, pp. 19-22

Cited by Sources: