Comptes Rendus
no. 11
Geometry/Functional Analysis
The Knaster problem and the geometry of high-dimensional cubes
[Le problème de Knaster et la géométrie des cubes en grande dimension]

Présenté par : Mikhaël Gromov

Boris S. Kashin1 ; Stanislaw J. Szarek2, 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
Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 931-936.

Nous étudions des questions du type suivant : Soit 𝒢 une matrice positive semi-définie, existe-t-il une suite de vecteurs dans ℝ 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.

We study questions of the following type: Given positive semi-definite matrix 𝒢, does there exist a sequence of vectors in ℝ 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00226-7
Boris S. Kashin 1 ; Stanislaw J. Szarek 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 
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Boris S. Kashin; Stanislaw J. Szarek. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00226-7/

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

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

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

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

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

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

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

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

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

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