Aleksandrov problem for isometric mappings

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