  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.
