The theory of
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
in the framework of
Banach spaces
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
has its beginning in a paper by
Stanisław Mazur
Stanisław Mieczysław Mazur (1 January 1905, Lwów – 5 November 1981, Warsaw) was a Polish mathematician and a member of the Polish Academy of Sciences.
Mazur made important contributions to geometrical methods in linear and nonlinear functio ...
and
Stanisław M. Ulam in 1932. They proved the
Mazur–Ulam theorem In mathematics, the Mazur–Ulam theorem states that if V and W are normed spaces over R and the mapping
:f\colon V\to W
is a surjective isometry, then f is affine. It was proved by Stanisław Mazur and Stanisław Ulam in response to a questi ...
stating that every isometry of a
normed real linear space onto a normed real linear space is a
linear mapping
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
up to translation. In 1970,
Aleksandr Danilovich Aleksandrov
Aleksandr Danilovich Aleksandrov (russian: Алекса́ндр Дани́лович Алекса́ндров, alternative transliterations: ''Alexandr'' or ''Alexander'' (first name), and ''Alexandrov'' (last name)) (4 August 1912 – 27 July 19 ...
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 and are normed linear spaces and if is a continuous and/or surjective mapping such that whenever vectors and in satisfy , then (the distance one preserving property or DOPP), is then necessarily an isometry?
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.
{{DEFAULTSORT:Aleksandrov-Rassias problem
Mathematical analysis
Metric geometry
Functional equations