One proves that the Mazur‐Ulam theorem can be extended in the framework of metric spaces us 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 alln × ndimensional positive definite matrices, appear as symmetric cones attached to Euclidean Jordan algebras. It turns out that the Mazur‐Ulam spaces provide a framework for new generalizations of the concept of convex function. © 2004 American Institute of Physics