  In this paper Hilbert spaces are characterized among Banach spaces in terms
of transitivity with respect to nicely behaved subgroups of the isometry group.
For example, the following result is typical here: If X is a real Banach space
isomorphic to a Hilbert space and convex-transitive with respect to the
isometric finite-dimensional perturbations of the identity, then X is already
isometric to a Hilbert space.
