On the isometric conjecture of Banach

Bor, Gil¹ ; Hernández Lamoneda, Luis¹ ; Jiménez-Desantiago, Valentín² ; Montejano, Luis²

¹ Centro de Investigación en Matemáticas, Guanajuato, Mexico
² Instituto de Matemáticas, Universidad Nacional Autónoma de México, Juriquilla, Mexico

Geometry & topology, Tome 25 (2021) no. 5, pp. 2621-2642.

Résumé

Let $V$ be a Banach space all of whose subspaces of a fixed dimension $n$ are isometric, with $1<n<dim\left(V\right)$. In 1932, S Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. In 1967, M Gromov answered it positively for even $n$. We give a positive answer for real $V$ and odd $n$ of the form $n=4k+1$, with the possible exception of $n=133$. Our proof relies on a new characterization of ellipsoids in ${ℝ}^n$ for $n\ge 5$, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.

Détail

DOI : 10.2140/gt.2021.25.2621

Classification : 52A21, 46B04
Keywords: convex body of revolution, structure group reduction

Bibliographie

[1] J F Adams, Lectures on Lie groups, Benjamin (1969)

[2] H Auerbach, S Mazur, S Ulam, Sur une propriété caractéristique de l’ellipsoïde, Monatsh. Math. Phys. 42 (1935) 45 | DOI

[3] S Banach, Théorie des opérations linéaires, 1, PWN (1932)

[4] A L Besse, Einstein manifolds, 10, Springer (1987) | DOI

[5] M Čadek, M Crabb, G–structures on spheres, Proc. Lond. Math. Soc. 93 (2006) 791 | DOI

[6] A Dvoretzky, A theorem on convex bodies and applications to Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 45 (1959) 223 | DOI

[7] A Dvoretzky, Some results on convex bodies and Banach spaces, from: "Proceedings of the international symposium on linear spaces", Jerusalem Acad. Press (1961) 123

[8] E B Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sb. 30(72) (1952) 349

[9] M L Gromov, A geometrical conjecture of Banach, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967) 1105

[10] D Husemoller, Fibre bundles, 20, Springer (1994) | DOI

[11] K Itô, editor, Encyclopedic dictionary of mathematics, MIT Press (1987)

[12] S Kobayashi, K Nomizu, Foundations of differential geometry, I, Interscience (1963)

[13] P Leonard, G–structures on spheres, Trans. Amer. Math. Soc. 157 (1971) 311 | DOI

[14] P Mani, Fields of planar bodies tangent to spheres, Monatsh. Math. 74 (1970) 145 | DOI

[15] H Martini, L Montejano, D Oliveros, Bodies of constant width: an introduction to convex geometry with applications, Birkhäuser (2019) | DOI

[16] V D Milman, New proof of the theorem of A Dvoretzky on intersections of convex bodies, Funkcional. Anal. i Priložen. 5 (1971) 28

[17] L Montejano, Convex bodies with homothetic sections, Bull. Lond. Math. Soc. 23 (1991) 381 | DOI

[18] L Montejano, Two applications of topology to convex geometry, Tr. Mat. Inst. Steklova 247 (2004) 182

[19] D Montgomery, H Samelson, Transformation groups of spheres, Ann. of Math. 44 (1943) 454 | DOI

[20] A L Onishchik, Topology of transitive transformation groups, Barth (1994)

[21] A Pełciński, On some problems of Banach, Wiadom. Mat. (2) 15 (1972) 3

[22] V Soltan, Characteristic properties of ellipsoids and convex quadrics, Aequationes Math. 93 (2019) 371 | DOI

[23] N Steenrod, The topology of fibre bundles, 14, Princeton Univ. Press (1951)

[24] G W Whitehead, Elements of homotopy theory, 61, Springer (1978) | DOI