The Knaster problem and the geometry of high-dimensional cubes

Boris S. Kashin; Stanislaw J. Szarek

doi:10.1016/S1631-073X(03)00226-7

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. 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

Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 931-936.

Abstract (VO)
Résumé

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.

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.

Reçu le : 2003-03-20
Accepté le : 2003-04-22
Publié le : 2003-06-01

DOI : 10.1016/S1631-073X(03)00226-7

Affiliations des auteurs :

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/

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