Isometric reflections on Banach spaces after a paper of A. Skorik and M. Zaidenberg

Let E be a real Banach space. A noon-one element e in E is said to be an is ometric reflection vector if there exist a maximal subspace M of E and a li near isometry F : E --> E fixing the elements of hi and satisfying F(e) = - e. We prove that each of the conditions (i) and (ii) below implies that E i s a Hilbert space. (i) There exists a nonrare subset of the unit sphere of E consisting only of isometric reflection vectors, (ii) There is an isometr ic reflection vector in E, the norm of E is convex transitive, and the iden tity component of the group of all surjective linear isometries on E relati ve to the strong operator topology is not reduced to the identity operator on E.