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The Aleksandrov-Benz-Rassias problem on linear \(n\)-normed spaces. (English) Zbl 1354.46014

Summary: This paper generalizes the Aleksandrov-Benz-Rassias problem on \(n\)-normed spaces.

MSC:

46B04 Isometric theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
51K05 General theory of distance geometry
Full Text: DOI

References:

[1] Mazur, S., Ulam, S.: Sur les transformationes isométriques d’espaces vectoriels normés. C. R. Acad. Sci. Paris 194, 946-948 (1932) · JFM 58.0423.01
[2] Aleksandrov, A.D.: Mappings of families of sets. Soviet Math. Dokl. 11, 116-120 (1970) · Zbl 0213.48903
[3] Benz, W.: A contribution to a theorem of Ulam and Mazur. Aequationes Math. 34, 61-63 (1987) · Zbl 0651.46022 · doi:10.1007/BF01840123
[4] Rassias, ThM, Šemrl, P.: On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings. Proc. Am. Math. Soc. 132, 919-925 (1993) · Zbl 0780.51010 · doi:10.1090/S0002-9939-1993-1111437-6
[5] Rassias, ThM: Is a distance one preserving mapping between metric spaces always an isometry? Am. Math. Mon. 90, 200 (1983) · Zbl 0512.54017 · doi:10.2307/2975550
[6] Mielnik, B., Rassias, ThM: On the Aleksandrov problem of conservative distances. Proc. Am. Math. Soc. 116, 1115-1118 (1992) · Zbl 0769.51005 · doi:10.1090/S0002-9939-1992-1101989-3
[7] Rassias, ThM, Sharma, C.S.: Properties of isometries. J. Nat. Geom. 3, 1-38 (1993) · Zbl 0776.51015
[8] Rassias, ThM: Properties of isometric mappings. J. Math. Anal. Appl. 235(1), 108-121 (1999) · Zbl 0936.46009 · doi:10.1006/jmaa.1999.6363
[9] Rassias, Th.M., Xiang, S.: On mappings with conservative distances and the Mazur-Ulam theorem. Publ. Fac. Electr. Eng. Univ. Belgrade Ser Math. 11, 1-8 (2000) · Zbl 1004.46007
[10] Rassias, Th.M, Xiang, S.: On Mazur-Ulam theorem and mappings which preserve distances. Nonlinear Funct. Anal. Appl. 5(2), 61-66 (2000) · Zbl 0981.46008
[11] Rassias, ThM: Isometries and approximate isometries. Int. J. Math. Math. Sci. 25, 73-91 (2001) · Zbl 0990.46004 · doi:10.1155/S0161171201004392
[12] Jung, S.M., Rassias, ThM: On distance-preserving mappings. J. Korean Math. Soc. 41(4), 667-680 (2004) · Zbl 1053.51008 · doi:10.4134/JKMS.2004.41.4.667
[13] Jung, S.M., Rassias, ThM: Mappings preserving two distances. Nonlinear Funct. Anal. Appl. 10(5), 717-723 (2005) · Zbl 1135.51013
[14] Ma, Y.: The Aleksandrov problem for unit distance preserving mapping. Acta Math. Sci. Ser. B Engl. Ed. 20, 359-364 (2000) · Zbl 0973.46011
[15] Gao, J.: On the Aleksandrov problem of distance preserving mapping. J. Math. Anal. Appl. 352, 583-590 (2009) · Zbl 1160.46004 · doi:10.1016/j.jmaa.2008.10.022
[16] Jing, Y.: The Aleksandrov problem in \[p\] p-normed spaces \[(0 < p \le 1)\](0<p≤1). Acta Sci. Nat. Univ. Nankai. 4, 91-96 (2008) · Zbl 1199.46042
[17] Chu, H., Lee, K., Park, C.: On the Aleksandrov problem in linear \[n\] n-normed spaces. Nonlinear Anal. 59, 1001-1011 (2004) · Zbl 1066.46005
[18] Chu, H., Choi, S., Kang, D.: Mapping of conservative distance in linear \[n\] n-normed spaces. Nonlinear Anal. 70, 1168-1174 (2009) · Zbl 1170.46022 · doi:10.1016/j.na.2008.02.002
[19] Park, C., Rassias, T.M.: Isometries on linear \[n\] n-normed spaces. JIPAM. J. Inequal. Pure Appl. Math. 7, 1-17 (2006) · Zbl 1137.46005
[20] Park, C., Alaca, C.: A new version of Mazur-Ulam theorem under weaker conditions in linear n-normed spaces. J. Comput. Anal. Appl. 16, 827-832 (2014) · Zbl 1308.46017
[21] Yunchen, X., Meimei, S.: Characterizations on isometries in linear \[n\] n-normed spaces. Nonlinear Anal. 72, 1895-1901 (2010) · Zbl 1196.46011 · doi:10.1016/j.na.2009.09.029
[22] Ma, Y.: On the Aleksandrov-Rassias problems on Linear n-normed spaces. J. Funct. Spaces Appl., Article ID 394216, 1-7 (2013) · Zbl 1283.46010
[23] Ma, Y.: The Aleksandrov problem and the Mazue-Ulam Theorem on Linear n-normed space. Bull. Korean Math. Soc. 50, 1631-1637 (2013) · Zbl 1292.46006 · doi:10.4134/BKMS.2013.50.5.1631
[24] Ma, Y.: Isometries on linear n-normed spaces. Ann. Acad. Sci. Fenn. Math 39, 973-981 (2014) · Zbl 1310.46013 · doi:10.5186/aasfm.2014.3941
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