On 2-local nonlinear surjective isometries on normed spaces and C^-algebras

We prove that if the closed unit ball of a normed space X has sufficiently many extreme points, then every mapping \Phi from X into itself with the following property is affine: For any pair of points in X, there exists a (not necessarily linear) surjective isometry on X that coincides with \Phi at the two points. We also consider surjectivity of such a mapping in some special cases including C ^*-algebras.