Aleksandrov problem for isometric mappings

Let , be two metric spaces, with respective distances , (cf. also Metric space). A mapping is defined to be an isometry if for all . A mapping is said to preserve the distance if for all with one has . A.D. Aleksandrov has posed the problem whether the existence of a single preserved distance for some mapping implies that is an isometry (cf. [a1]).

Even if , are normed vector spaces, the above problem is not easy to answer (note that, in this case, taking is no loss of generality). For example, the following question has not been solved yet (as of 2000): Is a mapping from to 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 : For , the answer is positive [a2], while for , 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 and are strictly convex vector spaces, provided is a homeomorphism and the dimension of is greater than (cf. [a12], [a20]).

The Aleksandrov problem has also been solved in some special cases of mappings which preserve two distances with an integer ratio greater than , with and strictly convex vector spaces and the dimension of greater than (cf. [a3], [a16], [a17]). Furthermore, when , are Hilbert spaces and the dimension of is greater than , a lot of work has been done (cf. [a21], [a22], [a23]; for example, when preserves the two distances and ; when and preserves and , then 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