Directionally Euclidean structures of Banach spaces

Volume 202 / 2011

Jarno Talponen Studia Mathematica 202 (2011), 191-203 MSC: Primary 46B07, 46C15; Secondary 43A35, 52A23. DOI: 10.4064/sm202-2-5

Abstract

We study Banach spaces with directionally asymptotically controlled ellipsoid-approximations of the unit ball in finite-dimensional sections. Here these ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids by means of a given functional of the dual space. The term `asymptotical' refers to the fact that we take `$\limsup$' over finite-dimensional subspaces.

This leads to isomorphic and isometric characterizations of Hilbert spaces. An application involving Mazur's rotation problem is given. We also discuss the stability of the family of ellipsoids as the dimension and geometry vary. The methods exploit ultrafilter techniques and we also apply them in conjunction with finite Auerbach bases to study the convexity properties of the duality mappings.

Authors

  • Jarno TalponenInstitute of Mathematics
    Aalto University
    P.O. Box 11100
    FI-00076 Aalto, Finland
    e-mail

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