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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a1301401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a1301402.png" /> be two metric spaces, with respective distances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a1301403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a1301404.png" /> (cf. also [[Metric space|Metric space]]). A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a1301405.png" /> is defined to be an isometry if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a1301406.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a1301407.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a1301408.png" /> is said to preserve the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a1301409.png" /> if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014010.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014011.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014012.png" />. A.D. Aleksandrov has posed the problem whether the existence of a single preserved distance for some mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014013.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014014.png" /> is an isometry (cf. [[#References|[a1]]]).
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Even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014016.png" /> are normed vector spaces, the above problem is not easy to answer (note that, in this case, taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014017.png" /> is no loss of generality). For example, the following question has not been solved yet (as of 2000): Is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014018.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014019.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014020.png" /> preserving unit distance necessarily an isometry (cf. [[#References|[a17]]])?
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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]]]).
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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 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014021.png" />: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014022.png" />, the answer is positive [[#References|[a2]]], while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014023.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014025.png" /> are strictly convex vector spaces, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014026.png" /> is a [[Homeomorphism|homeomorphism]] and the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014027.png" /> is greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014028.png" /> (cf. [[#References|[a12]]], [[#References|[a20]]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014029.png" /> which preserve two distances with an integer ratio greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014030.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014032.png" /> strictly convex vector spaces and the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014033.png" /> greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014034.png" /> (cf. [[#References|[a3]]], [[#References|[a16]]], [[#References|[a17]]]). Furthermore, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014036.png" /> are Hilbert spaces and the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014037.png" /> is greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014038.png" />, a lot of work has been done (cf. [[#References|[a21]]], [[#References|[a22]]], [[#References|[a23]]]; for example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014039.png" /> preserves the two distances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014041.png" />; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014043.png" /> preserves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014045.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130140/a13014046.png" /> is an affine isometry, etc.).
+
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]]].
 
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====
 
====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>
+
<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>

Latest revision as of 16:46, 1 July 2020

Let $X$, $Y$ be two metric spaces, with respective distances $d_1$, $d _ { 2 }$ (cf. also 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. [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. [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. [a14], [a15], [a16], [a18], [a17]).

The Aleksandrov problem has been solved for Euclidean spaces $X = Y = \mathbf{R} ^ { n }$: For $2 \leq n < \infty$, the answer is positive [a2], while for $n = 1 , \infty$, the answer is negative [a2], [a5], [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 and the dimension of $X$ is greater than $2$ (cf. [a12], [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. [a3], [a16], [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. [a21], [a22], [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 [a4], [a5], [a6], [a7], [a8], [a9], [a10], [a11].

References

[a1] A.D. Aleksandrov, "Mapping of families of sets" Soviet Math. Dokl. , 11 (1970) pp. 116–120
[a2] F.S. Beckman, D.A. Quarles, "On isometries of Euclidean spaces" Proc. Amer. Math. Soc. , 4 (1953) pp. 810–815
[a3] W. Benz, H. Berens, "A contribution to a theorem of Ulam–Mazur" Aquat. Math. , 34 (1987) pp. 61–63
[a4] D.G. Bourgain, "Approximate isometries" Bull. Amer. Math. Soc. , 52 (1946) pp. 704–714
[a5] K. Ciesielski, Th.M. Rassias, "On some properties of isometric mappings" Facta Univ. Ser. Math. Inform. , 7 (1992) pp. 107–115
[a6] G. Dolinar, "Generalized stability of isometries" J. Math. Anal. Appl. , 202 (2000) pp. 39–56
[a7] J. Gevirtz, "Stability of isometries on Banach spaces" Proc. Amer. Math. Soc. , 89 (1983) pp. 633–636
[a8] P.M. Gruber, "Stability of isometries" Trans. Amer. Math. Soc. , 245 (1978) pp. 263–277
[a9] J. Lindenstrauss, A. Szankowski, "Non linear perturbations of isometries" Astérisque , 131 (1985) pp. 357–371
[a10] D.H. Hyers, S.M. Mazur, "On approximate isometries" Bull. Amer. Math. Soc. , 51 (1945) pp. 288–292
[a11] S.M. Mastir, S. Ulam, "Sur les transformations isométriques d'espaces vectoriels normés" C.R. Acad. Sci. Paris , 194 (1932) pp. 946–948
[a12] B. Mielnik, Th.M. Rassias, "On the Aleksandrov problem of conservative distances" Proc. Amer. Math. Soc. , 116 (1992) pp. 1115–1118
[a13] Th.M. Rassias, "Some remarks on isometric mappings" Facta Univ. Ser. Math. Inform. , 2 (1987) pp. 49–52
[a14] Th.M. Rassias, "Is a distance one preserving maping between metric space always an isometry?" Amer. Math. Monthly , 90 (1983) pp. 200
[a15] 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
[a16] Th.M. Rassias, "Mappings that preserve unit distance" Indian J. Math. , 32 (1990) pp. 275–278
[a17] Th.M. Rassias, "Properties of isometries and approximate isometries" , Recent Progress in Inequalities , Kluwer Acad. Publ. (1998) pp. 325–345
[a18] Th.M. Rassias, "Remarks and problems" Aequat. Math. , 39 (1990) pp. 304
[a19] Th.M. Rassias, "Remarks and problems" Aequat. Math. , 56 (1998) pp. 304–306
[a20] 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
[a21] Th.M. Rassias, Shuhuang Xiang, "On mappings with conservative distance and the Mazur–Ulam theorem" Publ. EPT. , to appear (2000)
[a22] Shuhuang Xiang, "Aleksandrov problem and mappings which preserves distances" , Funct. Equations and Inequalities , Kluwer Acad. Publ. (2000) pp. 297–323
[a23] Shuhuang Xiang, "Mappings of conservative distances and the Mazur–Ulam theorem" J. Math. Anal. Appl. , 254 (2001) pp. 262–274
How to Cite This Entry:
Aleksandrov problem for isometric mappings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aleksandrov_problem_for_isometric_mappings&oldid=50017
This article was adapted from an original article by Shuhuang Xiang (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article