An Extension of the Mazur-ulam Theorem

One proves that the Mazur-Ulam theorem can be extended in the framework of metric spaces as long as a well behaved concept of midpoint is available. This leads to the new concept of Mazur-Ulam space. Besides the classical case of real normed spaces, other examples such as Sym++(n,R), the space of all n × n dimensional positive definite matrices, appear as cones attached to suitable Euclidean Jordan algebras. It turns out that the MazurUlam spaces provide a framework for new generalizations of the concept of convex function.