Convex-transitive characterizations of Hilbert spaces

TL;DR

This paper characterizes when certain convex-transitive Banach spaces with specific projections are actually Hilbert spaces, advancing understanding of the Banach-Mazur rotation problem and related questions.

Contribution

It provides new conditions involving weak topology and norm geometry that ensure convex-transitive spaces are Hilbert spaces.

Findings

Convex-transitive Banach spaces with a 1-dimensional bicontractive projection are Hilbert spaces under mild conditions.

The results offer partial solutions to the Banach-Mazur rotation problem.

Addresses a question by B. Randrianantoanina on convex-transitive spaces.

Abstract

In this paper we investigate real convex-transitive Banach spaces X, which admit a 1-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. The results obtained can be regarded as partial answers to the well-known Banach-Mazur rotation problem, as well as to a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces.