Directionally Euclidean structures of Banach spaces

Jarno Talponen. "Directionally Euclidean structures of Banach spaces." Studia Mathematica 202.2 (2011): 191-203. <http://eudml.org/doc/285853>.

@article{JarnoTalponen2011,
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 'lim sup' 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. },
author = {Jarno Talponen},
journal = {Studia Mathematica},
keywords = {Auerbach basis; minimum volume ellipsoid; Hilbert space},
language = {eng},
number = {2},
pages = {191-203},
title = {Directionally Euclidean structures of Banach spaces},
url = {http://eudml.org/doc/285853},
volume = {202},
year = {2011},
}

TY - JOUR
AU - Jarno Talponen
TI - Directionally Euclidean structures of Banach spaces
JO - Studia Mathematica
PY - 2011
VL - 202
IS - 2
SP - 191
EP - 203
AB - 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 'lim sup' 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.
LA - eng
KW - Auerbach basis; minimum volume ellipsoid; Hilbert space
UR - http://eudml.org/doc/285853
ER -