View source for Aleksandrov problem for isometric mappings

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]]]).

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]]])?

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]]]).

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.).

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>