Directionally Euclidean Structures of Banach Spaces

TL;DR

This paper investigates the geometric structure of Banach spaces through directionally controlled ellipsoids, providing characterizations of Hilbert spaces and insights into their asymptotic geometric properties.

Contribution

It introduces a novel approach using directionally asymptotically controlled ellipsoids to characterize Hilbert spaces and analyze their geometric complexity.

Findings

Provides isomorphic and isometric characterizations of Hilbert spaces.

Analyzes the complexity of ellipsoid families as dimension and geometry vary.

Includes an application to Mazur's rotation problem.

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.