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.^[1] They proved the Mazur–Ulam theorem stating that every isometry of a normed real linear space onto a normed real linear space is a linear mapping up to translation. In 1970, Aleksandr Danilovich Aleksandrov asked whether the existence of a single conservative distance for a mapping implies that it is an isometry. Themistocles M. Rassias posed the following problem:

Aleksandrov–Rassias Problem. 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 ${\displaystyle \lVert x-y\rVert =1}$, then ${\displaystyle \lVert T(X)-T(Y)\rVert =1}$ (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.

References

• P. M. Pardalos, P. G. Georgiev and H. M. Srivastava (eds.), Nonlinear Analysis. Stability, Approximation, and Inequalities. In honor of Themistocles M. Rassias on the occasion of his 60th birthday, Springer, New York, 2012.
• A. D. Aleksandrov, Mapping of families of sets, Soviet Math. Dokl. 11(1970), 116–120.
• On the Aleksandrov-Rassias problem for isometric mappings
• On the Aleksandrov-Rassias problem and the geometric invariance in Hilbert spaces
• S.-M. Jung and K.-S. Lee, An inequality for distances between 2n points and the Aleksandrov–Rassias problem, J. Math. Anal. Appl. 324(2)(2006), 1363–1369.
• S. Xiang, Mappings of conservative distances and the Mazur–Ulam theorem, J. Math. Anal. Appl. 254(1)(2001), 262–274.
• S. Xiang, On the Aleksandrov problem and Rassias problem for isometric mappings, Nonlinear Functional Analysis and Appls. 6(2001), 69-77.
• S. Xiang, On approximate isometries, in : Mathematics in the 21st Century (eds. K. K. Dewan and M. Mustafa), Deep Publs. Ltd., New Delhi, 2004, pp. 198–210.