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Let $X$, $Y$ be two metric spaces, with respective distances $d_1$, $d _ { 2 }$ (cf. also [[Metric space|Metric space]]). A mapping $f : X \rightarrow Y$ is defined to be an isometry if $d_{2} ( f ( x ) , f ( y ) ) = d _ { 1 } ( x , y )$ for all $x , y \in X$. A mapping $f : X \rightarrow Y$ is said to preserve the distance $r$ if for all $x , y \in X$ with $d _ { 1 } ( x , y ) = r$ one has $d _ { 2 } ( f ( x ) , f ( y ) ) = r$. A.D. Aleksandrov has posed the problem whether the existence of a single preserved distance for some mapping $f$ implies that $f$ is an isometry (cf. [[#References|[a1]]]).

Even if $X$, $Y$ are normed vector spaces, the above problem is not easy to answer (note that, in this case, taking $r = 1$ is no loss of generality). For example, the following question has not been solved yet (as of 2000): Is a mapping $f$ from $\mathbf{R} ^ { 2 }$ to $\mathbf{R} ^ { 3 }$ preserving unit distance necessarily an isometry (cf. [[#References|[a17]]])?

The discussion of Aleksandrov's problem under certain additional conditions on the given mapping preserving unit distance has led to several interesting and new problems (cf. [[#References|[a14]]], [[#References|[a15]]], [[#References|[a16]]], [[#References|[a18]]], [[#References|[a17]]]).

The Aleksandrov problem has been solved for Euclidean spaces $X = Y = \mathbf{R} ^ { n }$: For $2 \leq n &lt; \infty$, the answer is positive [[#References|[a2]]], while for $n = 1 , \infty$, the answer is negative [[#References|[a2]]], [[#References|[a5]]], [[#References|[a13]]]. In general normed vector spaces, the answer is positive for a mapping that is contractive, surjective and preserves the unit distance. The problem also has a positive solution when $X$ and $Y$ are strictly convex vector spaces, provided $f$ is a [[Homeomorphism|homeomorphism]] and the dimension of $X$ is greater than $2$ (cf. [[#References|[a12]]], [[#References|[a20]]]).

The Aleksandrov problem has also been solved in some special cases of mappings $f : X \rightarrow Y$ which preserve two distances with an integer ratio greater than $1$, with $X$ and $Y$ strictly convex vector spaces and the dimension of $X$ greater than $1$ (cf. [[#References|[a3]]], [[#References|[a16]]], [[#References|[a17]]]). Furthermore, when $X$, $Y$ are Hilbert spaces and the dimension of $X$ is greater than $1$, a lot of work has been done (cf. [[#References|[a21]]], [[#References|[a22]]], [[#References|[a23]]]; for example, when $f$ preserves the two distances $1$ and $\sqrt { 3 }$; when $\operatorname{dim} X \geq 3$ and $f$ preserves $1$ and $\sqrt { 2 }$, then $f$ is an affine isometry, etc.).

Problems connected with stability of isometries as well as non-linear perturbations of isometries have been extensively studied in [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  A.D. Aleksandrov,   "Mapping of families of sets"  ''Soviet Math. Dokl.'' , '''11'''  (1970)  pp. 116–120</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  F.S. Beckman,   D.A. Quarles,   "On isometries of Euclidean spaces"  ''Proc. Amer. Math. Soc.'' , '''4'''  (1953)  pp. 810–815</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  W. Benz,   H. Berens,   "A contribution to a theorem of Ulam–Mazur"  ''Aquat. Math.'' , '''34'''  (1987)  pp. 61–63</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  D.G. Bourgain,   "Approximate isometries"  ''Bull. Amer. Math. Soc.'' , '''52'''  (1946)  pp. 704–714</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  K. Ciesielski,   Th.M. Rassias,   "On some properties of isometric mappings"  ''Facta Univ. Ser. Math. Inform.'' , '''7'''  (1992)  pp. 107–115</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  G. Dolinar,   "Generalized stability of isometries"  ''J. Math. Anal. Appl.'' , '''202'''  (2000)  pp. 39–56</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J. Gevirtz,   "Stability of isometries on Banach spaces"  ''Proc. Amer. Math. Soc.'' , '''89'''  (1983)  pp. 633–636</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  P.M. Gruber,   "Stability of isometries"  ''Trans. Amer. Math. Soc.'' , '''245'''  (1978)  pp. 263–277</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  J. Lindenstrauss,   A. Szankowski,   "Non linear perturbations of isometries"  ''Astérisque'' , '''131'''  (1985)  pp. 357–371</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  D.H. Hyers,   S.M. Mazur,   "On approximate isometries"  ''Bull. Amer. Math. Soc.'' , '''51'''  (1945)  pp. 288–292</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  S.M. Mastir,   S. Ulam,   "Sur les transformations isométriques d'espaces vectoriels normés"  ''C.R. Acad. Sci. Paris'' , '''194'''  (1932)  pp. 946–948</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  B. Mielnik,   Th.M. Rassias,   "On the Aleksandrov problem of conservative distances"  ''Proc. Amer. Math. Soc.'' , '''116'''  (1992)  pp. 1115–1118</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  Th.M. Rassias,   "Some remarks on isometric mappings"  ''Facta Univ. Ser. Math. Inform.'' , '''2'''  (1987)  pp. 49–52</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  Th.M. Rassias,   "Is a distance one preserving maping between metric space always an isometry?"  ''Amer. Math. Monthly'' , '''90'''  (1983)  pp. 200</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  Th.M. Rassias,   "The stability of linear mappings and some problems on isometries" , ''Proc. Internat. Conf. Math. Anal. Appl. Kuwait, 1985'' , Pergamon  (1988)  pp. 175–184</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  Th.M. Rassias,   "Mappings that preserve unit distance"  ''Indian J. Math.'' , '''32'''  (1990)  pp. 275–278</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  Th.M. Rassias,   "Properties of isometries and approximate isometries" , ''Recent Progress in Inequalities'' , Kluwer Acad. Publ.  (1998)  pp. 325–345</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  Th.M. Rassias,   "Remarks and problems"  ''Aequat. Math.'' , '''39'''  (1990)  pp. 304</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  Th.M. Rassias,   "Remarks and problems"  ''Aequat. Math.'' , '''56'''  (1998)  pp. 304–306</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  Th.M. Rassias,   P. Semrl,   "On the Masur–Ulam theorem and the Aleksandrov probiem for unit distance preserving mapping"  ''Proc. Amer. Math. Soc.'' , '''118'''  (1993)  pp. 919–925</td></tr><tr><td valign="top">[a21]</td> <td valign="top">  Th.M. Rassias,   Shuhuang Xiang,   "On mappings with conservative distance and the Mazur–Ulam theorem"  ''Publ. EPT.'' , '''to appear'''  (2000)</td></tr><tr><td valign="top">[a22]</td> <td valign="top">  Shuhuang Xiang,   "Aleksandrov problem and mappings which preserves distances" , ''Funct. Equations and Inequalities'' , Kluwer Acad. Publ.  (2000)  pp. 297–323</td></tr><tr><td valign="top">[a23]</td> <td valign="top">  Shuhuang Xiang,   "Mappings of conservative distances and the Mazur–Ulam theorem"  ''J. Math. Anal. Appl.'' , '''254'''  (2001)  pp. 262–274</td></tr></table>