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 (, ) 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 vectors and ...
has its beginning in a paper by
Stanisław Mazur
Stanisław Mieczysław Mazur (; 1 January 1905 – 5 November 1981) was a Polish mathematician and a member of the Polish Academy of Sciences.
Mazur made important contributions to geometrical methods in linear and nonlinear functional analysis ...
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 Map (mathematics), mapping V \to W between two vec ...
up to translation. In 1970,
Aleksandr Danilovich Aleksandrov
Aleksandr Danilovich Aleksandrov (; 4 August 1912 – 27 July 1999) was a Soviet and Russian mathematician, physicist, philosopher and mountaineer.
Personal life
Aleksandr Aleksandrov was born in 1912 in Volyn, Ryazan Oblast. His father was ...
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 space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
s by the
Beckman–Quarles theorem.
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