This is an announcement for the paper "Directionally Euclidean structures of Banach spaces" by Jarno Talponen.
Abstract: We study spaces with directionally asymptotically controlled ellipsoids approximating the unit ball in finite-dimensions. 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 with 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 some isomorphic and isometric characterizations of Hilbert spaces. An application involving Mazur's rotation problem is given. We also discuss the complexity of the family of ellipsoids as the dimension and geometry vary.
Archive classification: math.FA
Mathematics Subject Classification: Primary 46C15, Secondary 52A23
Remarks: 10 pages
Submitted from: talponen@cc.hut.fi
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1005.2737
or