Mazur rotation problem

Mazur's rotation problem asks whether transitivity of the isometry group on the unit sphere characterizes Hilbert space among separable Banach spaces: if a separable real Banach space $X$ has the property that for any norm-one vectors $x,y$ there is a surjective linear isometry of $X$ taking $x$ to $y$, must $X$ be linearly isometric to a Hilbert space? The problem dates to 1932 and the circle around Banach's monograph, where it is recorded [Banach1932OperationsLineaires].

Hilbert spaces plainly have transitive spheres, and in finite dimensions transitivity does force the norm to be Euclidean. Separability is essential in the infinite-dimensional question, since nonseparable transitive spaces that are not Hilbertian are known. Affirmative answers have been obtained under additional hypotheses — smoothness conditions on the norm or strengthened forms of homogeneity beyond bare transitivity — as described in [CabelloSanchez2001MazurRotations], [BecerraRodriguez2002Transitivity].

In the separable infinite-dimensional case, however, transitivity of the unit sphere alone has not been shown to imply the parallelogram law, and the problem remains open.