The theory of isometries in the framework of Banach spaces has its beginning in a paper by Stanisław Mazur and Stanisław M. Ulam in 1932.
In 1970, Aleksandr Danilovich Aleksandrov asked whether the existence of a single distance that is preserved by a mapping implies that it is an isometry, as it does for Euclidean spaces by the Beckman–Quarles theorem.
If X and Y are normed linear spaces and if T : X → Y is a continuous and/or surjective mapping such that whenever vectors x and y in X satisfy
(the distance one preserving property or DOPP), is T then necessarily an isometry?
[2]There have been several attempts in the mathematical literature by a number of researchers for the solution to this problem.