Isometric reflection vectors in Banach spaces

The aim of this paper is to study the set IrX of isometric reflection vectors of a real Banach space X. We deal with geometry of isometric reflection vectors and parallelogram identity vectors, and we prove that a real Banach space is a Hilbert space if the set of parallelogram identity vectors has nonempty interior. It is also shown that every real Banach space can be decomposed as an I r -sum of a Hilbert space and a Banach space with some points which are not isometric reflection vectors. Finally, we give a new proof of the Becerra–Rodríguez result: a real Banach space X is a Hilbert space if and only if IrX is not rare.  2004 Elsevier Inc. All rights reserved.