ON THE MAZUR-ULAM THEOREM AND THE ALEKSANDROV PROBLEM FOR UNIT DISTANCE PRESERVING MAPPINGS

Let X and Y be two real normed vector spaces. A mapping f: X --> Y pre serves unit distance in both directions iff for all x, y is-an-element -of X with \\x - y\\ = 1 it follows that \\f(x) - f(y)\\ and conversel y. In this paper we shall study, instead of isometries, mappings satis fying the weaker assumption that they preserve unit distance in both d irections. We shall prove that such mappings are not very far from bei ng isometries. This problem was asked by A. D. Aleksandrov. The first classical result that characterizes isometries between normed real vec tor spaces goes back to S. Mazur and S. Ulam in 1932. We also obtain a n extension of the Mazur-Ulam theorem.