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Gao, Jinmei

On the Alexandrov problem of distance preserving mapping. (English) Zbl 1160.46004

J. Math. Anal. Appl. 352, No. 2, 583-590 (2009).
Summary: The author has studied the Alexandrov problem of area preserving mappings in linear 2-normed spaces and has provided some remarks for the generalization of earlier results of H.Y. Chu, C.G. Park and W.G. Park [J. Math. Anal. Appl. 289, No. 2, 666–672 (2004; Zbl 1045.46002)]. In addition the author has introduced the concept of linear \((2,p)\)-normed spaces and for such spaces he has solved the Alexandrov problem.

Cited in 2 Documents

MSC:

46A70 Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)
54A40 Fuzzy topology

Keywords:

linear 2-normed spaces; 2-Lipschitz mapping; 2-isometry; AOPP; linear \((2; p)\)-normed spaces

Citations:

Zbl 1045.46002
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Full Text: DOI

References:

[1] Alexandrov, A. D., Mappings of families of sets, Soviet Math., 11, 116-120 (1970) · Zbl 0213.48903
[2] Chu, H. Y.; Park, C. G.; Park, W. G., The Alexandrov problem in linear 2-normed spaces, J. Math. Anal. Appl., 289, 666-672 (2004) · Zbl 1045.46002
[3] Ding, G. G., On isometric extensions and distance one preserving mappings, Taiwanese J. Math., 10, 1, 243-249 (2006) · Zbl 1107.46008
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[5] Rassias, Th. M.; ��emrl, P., On the Mazur-Ulam problem and the Alexandrov problem for unit distance preserving mappings, Proc. Amer. Math. Soc., 118, 919-925 (1993) · Zbl 0780.51010
[6] Ma, Y. M.; Wang, J. Y., On the A.D. Alexandrov problem of isometric mapping, J. Math. Res. Exposition, 23, 4, 623-630 (2003) · Zbl 1159.46303
[7] Rassias, J. M.; Xiang, S.; Rassias, M. J., On the Alexandrov and triangle isometry Ulam stability problem, Int. J. Appl. Math. Stat., 7, 133-142 (2007)
[8] Mielnik, B.; Rassias, Th. M., On the Aleksandrov problem of conservative distances, Proc. Amer. Math. Soc., 116, 1115-1118 (1992) · Zbl 0769.51005
[9] White, A., 2-Banach spaces, Math. Nachr., 42, 43-60 (1969) · Zbl 0185.20003
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