# THE ALEKSANDROV PROBLEM AND THE MAZUR-ULAM THEOREM ON LINEAR n-NORMED SPACES

• Yumei, Ma (Department of Mathematics Dalian Nationality University)
• Published : 2013.09.30

#### Abstract

This paper generalizes the Aleksandrov problem and Mazur Ulam theorem to the case of $n$-normed spaces. For real $n$-normed spaces X and Y, we will prove that $f$ is an affine isometry when the mapping satisfies the weaker assumptions that preserves unit distance, $n$-colinear and 2-colinear on same-order.

#### References

1. A. D. Aleksandrov, Mappings of families of sets, Soviet Math. 11 (1970), 116-120.
2. H. Chu, S. Choi, and D. Kang, Mapping of conservative distance in linear n-normed spaces, Nonlinear Anal. 70 (2009), no. 3, 1168-1174. https://doi.org/10.1016/j.na.2008.02.002
3. H. Chu, K. Lee, and C. Park, On the Aleksandrov problem in linear n-normed spaces, Nonlinear Anal. 59 (2004), no. 7, 1001-1011.
4. H. Chu and C. Park, The Aleksandrov problem in linear 2-normed spaces, J. Math. Anal. Appl. 289 (2004), no. 2, 666-672. https://doi.org/10.1016/j.jmaa.2003.09.009
5. J. Gao, On the Aleksandrov problem of distance preserving mapping, J. Math. Anal. Appl. 352 (2009), 583-590. https://doi.org/10.1016/j.jmaa.2008.10.022
6. Y. Ma, The Aleksandrov problem for unit distance preserving mapping, Acta Math. Sci. Ser. B Engl. Ed. 20 (2000), no. 3, 359-364.
7. Y. Ma and J. Wang, Some researches about isometric mapping, J. Math. Res. Exposition. 4 (2003), 123-127.
8. S. Mazur and S. Ulam, Sur les transformationes isometriques d'espaces vectoriels normes, C. R. Acad. Sci. 194 (1932), 946-948.
9. T. M. Rassias, On the A. D. Aleksandrov problem of conservative distances and the Mazur-Ulam theorem, Nonlinear Anal. 47 (2001), 108-121.
10. T. M. Rassias and P. Semrl, On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings, Proc. Amer. Math. Soc. 132 (1993), no. 3, 919-925.

#### Cited by

1. The Aleksandrov–Benz–Rassias problem on linear n-normed spaces vol.180, pp.2, 2016, https://doi.org/10.1007/s00605-015-0786-8
2. Isometry on Linear n-G-quasi Normed Spaces vol.60, pp.02, 2017, https://doi.org/10.4153/CMB-2016-061-9